TPTP Problem File: SLH0342^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : SCC_Bloemen_Sequential/0000_SCC_Bloemen_Sequential/prob_02847_097991__6510412_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1480 ( 746 unt; 190 typ; 0 def)
% Number of atoms : 3427 (1465 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 11880 ( 362 ~; 57 |; 299 &;9959 @)
% ( 0 <=>;1203 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 6 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 682 ( 682 >; 0 *; 0 +; 0 <<)
% Number of symbols : 173 ( 170 usr; 18 con; 0-4 aty)
% Number of variables : 3761 ( 459 ^;3214 !; 88 ?;3761 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:54:19.873
%------------------------------------------------------------------------------
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% Relevant facts (1277)
thf(fact_0_dfs__dfss__rel_Ocong,axiom,
sCC_Bl907557413677168252_rel_v = sCC_Bl907557413677168252_rel_v ).
% dfs_dfss_rel.cong
thf(fact_1_vs__case,axiom,
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!= bot_bot_set_v ) ).
% vs_case
thf(fact_2_sub__env__trans,axiom,
! [E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ).
% sub_env_trans
thf(fact_3_predfss,axiom,
sCC_Bl1748261141445803503t_unit @ successors @ va @ ea ).
% predfss
thf(fact_4_w__def,axiom,
( w
= ( fChoice_v
@ ^ [X: v] : ( member_v @ X @ ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) ) ) ) ).
% w_def
thf(fact_5_some__equality,axiom,
! [P: v > $o,A: v] :
( ( P @ A )
=> ( ! [X2: v] :
( ( P @ X2 )
=> ( X2 = A ) )
=> ( ( fChoice_v @ P )
= A ) ) ) ).
% some_equality
thf(fact_6_some__eq__trivial,axiom,
! [X3: v] :
( ( fChoice_v
@ ^ [Y: v] : ( Y = X3 ) )
= X3 ) ).
% some_eq_trivial
thf(fact_7_some__sym__eq__trivial,axiom,
! [X3: v] :
( ( fChoice_v
@ ( ^ [Y2: v,Z: v] : ( Y2 = Z )
@ X3 ) )
= X3 ) ).
% some_sym_eq_trivial
thf(fact_8_dfss,axiom,
( ( sCC_Bloemen_dfss_v @ successors @ va @ ea )
= ( sCC_Bloemen_dfss_v @ successors @ va @ e ) ) ).
% dfss
thf(fact_9_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ ea ).
% local.wf
thf(fact_10__092_060open_062v_A_092_060notin_062_Aexplored_Ae_092_060close_062,axiom,
~ ( member_v @ va @ ( sCC_Bl157864678168468314t_unit @ ea ) ) ).
% \<open>v \<notin> explored e\<close>
thf(fact_11__092_060open_062v_A_092_060in_062_Avisited_Ae_092_060close_062,axiom,
member_v @ va @ ( sCC_Bl4645233313691564917t_unit @ ea ) ).
% \<open>v \<in> visited e\<close>
thf(fact_12_DiffI,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_13_DiffI,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ A2 )
=> ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ ( minus_minus_set_v @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_14_Diff__iff,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A2 )
& ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_15_Diff__iff,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B ) )
= ( ( member_v @ C @ A2 )
& ~ ( member_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_16_Diff__idemp,axiom,
! [A2: set_v,B: set_v] :
( ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ B ) @ B )
= ( minus_minus_set_v @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_17_empty__iff,axiom,
! [C: v] :
~ ( member_v @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_18_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_19_empty__iff,axiom,
! [C: set_v] :
~ ( member_set_v @ C @ bot_bot_set_set_v ) ).
% empty_iff
thf(fact_20_all__not__in__conv,axiom,
! [A2: set_v] :
( ( ! [X: v] :
~ ( member_v @ X @ A2 ) )
= ( A2 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_21_all__not__in__conv,axiom,
! [A2: set_Product_prod_v_v] :
( ( ! [X: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X @ A2 ) )
= ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_22_all__not__in__conv,axiom,
! [A2: set_set_v] :
( ( ! [X: set_v] :
~ ( member_set_v @ X @ A2 ) )
= ( A2 = bot_bot_set_set_v ) ) ).
% all_not_in_conv
thf(fact_23_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X: v] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_24_Collect__empty__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( ! [X: product_prod_v_v] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_25_Collect__empty__eq,axiom,
! [P: set_v > $o] :
( ( ( collect_set_v @ P )
= bot_bot_set_set_v )
= ( ! [X: set_v] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_26_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X: v] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_27_empty__Collect__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P ) )
= ( ! [X: product_prod_v_v] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_28_empty__Collect__eq,axiom,
! [P: set_v > $o] :
( ( bot_bot_set_set_v
= ( collect_set_v @ P ) )
= ( ! [X: set_v] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_29_Diff__cancel,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ A2 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_30_Diff__cancel,axiom,
! [A2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ A2 )
= bot_bot_set_set_v ) ).
% Diff_cancel
thf(fact_31_Diff__cancel,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ A2 @ A2 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_32_empty__Diff,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A2 )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_33_empty__Diff,axiom,
! [A2: set_set_v] :
( ( minus_7228012346218142266_set_v @ bot_bot_set_set_v @ A2 )
= bot_bot_set_set_v ) ).
% empty_Diff
thf(fact_34_empty__Diff,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A2 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_35_Diff__empty,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= A2 ) ).
% Diff_empty
thf(fact_36_Diff__empty,axiom,
! [A2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ bot_bot_set_set_v )
= A2 ) ).
% Diff_empty
thf(fact_37_Diff__empty,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ A2 @ bot_bot_set_v )
= A2 ) ).
% Diff_empty
thf(fact_38_pre__dfss__pre__dfs,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( sCC_Bl36166008131615352t_unit @ successors @ W @ E ) ) ) ) ).
% pre_dfss_pre_dfs
thf(fact_39_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_40_emptyE,axiom,
! [A: v] :
~ ( member_v @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_41_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_42_emptyE,axiom,
! [A: set_v] :
~ ( member_set_v @ A @ bot_bot_set_set_v ) ).
% emptyE
thf(fact_43_equals0D,axiom,
! [A2: set_v,A: v] :
( ( A2 = bot_bot_set_v )
=> ~ ( member_v @ A @ A2 ) ) ).
% equals0D
thf(fact_44_equals0D,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v] :
( ( A2 = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A @ A2 ) ) ).
% equals0D
thf(fact_45_equals0D,axiom,
! [A2: set_set_v,A: set_v] :
( ( A2 = bot_bot_set_set_v )
=> ~ ( member_set_v @ A @ A2 ) ) ).
% equals0D
thf(fact_46_equals0I,axiom,
! [A2: set_v] :
( ! [Y3: v] :
~ ( member_v @ Y3 @ A2 )
=> ( A2 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_47_equals0I,axiom,
! [A2: set_Product_prod_v_v] :
( ! [Y3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y3 @ A2 )
=> ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_48_equals0I,axiom,
! [A2: set_set_v] :
( ! [Y3: set_v] :
~ ( member_set_v @ Y3 @ A2 )
=> ( A2 = bot_bot_set_set_v ) ) ).
% equals0I
thf(fact_49_ex__in__conv,axiom,
! [A2: set_v] :
( ( ? [X: v] : ( member_v @ X @ A2 ) )
= ( A2 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_50_ex__in__conv,axiom,
! [A2: set_Product_prod_v_v] :
( ( ? [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ A2 ) )
= ( A2 != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_51_ex__in__conv,axiom,
! [A2: set_set_v] :
( ( ? [X: set_v] : ( member_set_v @ X @ A2 ) )
= ( A2 != bot_bot_set_set_v ) ) ).
% ex_in_conv
thf(fact_52_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_53_empty__def,axiom,
( bot_bot_set_v
= ( collect_v
@ ^ [X: v] : $false ) ) ).
% empty_def
thf(fact_54_empty__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] : $false ) ) ).
% empty_def
thf(fact_55_empty__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v
@ ^ [X: set_v] : $false ) ) ).
% empty_def
thf(fact_56_graph_Odfss_Ocong,axiom,
sCC_Bloemen_dfss_v = sCC_Bloemen_dfss_v ).
% graph.dfss.cong
thf(fact_57_minus__set__def,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( minus_9095120230875558447_v_v_o
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ A3 )
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_58_minus__set__def,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A3: set_set_v,B2: set_set_v] :
( collect_set_v
@ ( minus_minus_set_v_o
@ ^ [X: set_v] : ( member_set_v @ X @ A3 )
@ ^ [X: set_v] : ( member_set_v @ X @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_59_minus__set__def,axiom,
( minus_minus_set_v
= ( ^ [A3: set_v,B2: set_v] :
( collect_v
@ ( minus_minus_v_o
@ ^ [X: v] : ( member_v @ X @ A3 )
@ ^ [X: v] : ( member_v @ X @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_60_some__in__eq,axiom,
! [A2: set_v] :
( ( member_v
@ ( fChoice_v
@ ^ [X: v] : ( member_v @ X @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_v ) ) ).
% some_in_eq
thf(fact_61_some__in__eq,axiom,
! [A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v
@ ( fChoic927883735564035387od_v_v
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ A2 ) )
@ A2 )
= ( A2 != bot_bo723834152578015283od_v_v ) ) ).
% some_in_eq
thf(fact_62_some__in__eq,axiom,
! [A2: set_set_v] :
( ( member_set_v
@ ( fChoice_set_v
@ ^ [X: set_v] : ( member_set_v @ X @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_set_v ) ) ).
% some_in_eq
thf(fact_63_DiffD2,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ).
% DiffD2
thf(fact_64_DiffD2,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B ) )
=> ~ ( member_v @ C @ B ) ) ).
% DiffD2
thf(fact_65_DiffD1,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A2 ) ) ).
% DiffD1
thf(fact_66_DiffD1,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B ) )
=> ( member_v @ C @ A2 ) ) ).
% DiffD1
thf(fact_67_DiffE,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A2 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% DiffE
thf(fact_68_DiffE,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A2 @ B ) )
=> ~ ( ( member_v @ C @ A2 )
=> ( member_v @ C @ B ) ) ) ).
% DiffE
thf(fact_69_some__eq__imp,axiom,
! [P: v > $o,A: v,B3: v] :
( ( ( fChoice_v @ P )
= A )
=> ( ( P @ B3 )
=> ( P @ A ) ) ) ).
% some_eq_imp
thf(fact_70_tfl__some,axiom,
! [P2: v > $o,X4: v] :
( ( P2 @ X4 )
=> ( P2 @ ( fChoice_v @ P2 ) ) ) ).
% tfl_some
thf(fact_71_Eps__cong,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X2: v] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( fChoice_v @ P )
= ( fChoice_v @ Q ) ) ) ).
% Eps_cong
thf(fact_72_someI,axiom,
! [P: v > $o,X3: v] :
( ( P @ X3 )
=> ( P @ ( fChoice_v @ P ) ) ) ).
% someI
thf(fact_73_set__diff__eq,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
& ~ ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_74_set__diff__eq,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A3: set_set_v,B2: set_set_v] :
( collect_set_v
@ ^ [X: set_v] :
( ( member_set_v @ X @ A3 )
& ~ ( member_set_v @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_75_set__diff__eq,axiom,
( minus_minus_set_v
= ( ^ [A3: set_v,B2: set_v] :
( collect_v
@ ^ [X: v] :
( ( member_v @ X @ A3 )
& ~ ( member_v @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_76_some1__equality,axiom,
! [P: v > $o,A: v] :
( ? [X4: v] :
( ( P @ X4 )
& ! [Y3: v] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( ( P @ A )
=> ( ( fChoice_v @ P )
= A ) ) ) ).
% some1_equality
thf(fact_77_mem__Collect__eq,axiom,
! [A: v,P: v > $o] :
( ( member_v @ A @ ( collect_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_78_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_79_mem__Collect__eq,axiom,
! [A: set_v,P: set_v > $o] :
( ( member_set_v @ A @ ( collect_set_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A2: set_v] :
( ( collect_v
@ ^ [X: v] : ( member_v @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A2: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_set_v] :
( ( collect_set_v
@ ^ [X: set_v] : ( member_set_v @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83_Collect__cong,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X2: set_v] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_v @ P )
= ( collect_set_v @ Q ) ) ) ).
% Collect_cong
thf(fact_84_some__eq__ex,axiom,
! [P: v > $o] :
( ( P @ ( fChoice_v @ P ) )
= ( ? [X5: v] : ( P @ X5 ) ) ) ).
% some_eq_ex
thf(fact_85_someI2__bex,axiom,
! [A2: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ? [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X2: product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X2 @ A2 )
& ( P @ X2 ) )
=> ( Q @ X2 ) )
=> ( Q
@ ( fChoic927883735564035387od_v_v
@ ^ [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A2 )
& ( P @ X ) ) ) ) ) ) ).
% someI2_bex
thf(fact_86_someI2__bex,axiom,
! [A2: set_v,P: v > $o,Q: v > $o] :
( ? [X4: v] :
( ( member_v @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X2: v] :
( ( ( member_v @ X2 @ A2 )
& ( P @ X2 ) )
=> ( Q @ X2 ) )
=> ( Q
@ ( fChoice_v
@ ^ [X: v] :
( ( member_v @ X @ A2 )
& ( P @ X ) ) ) ) ) ) ).
% someI2_bex
thf(fact_87_someI2__ex,axiom,
! [P: v > $o,Q: v > $o] :
( ? [X_1: v] : ( P @ X_1 )
=> ( ! [X2: v] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( fChoice_v @ P ) ) ) ) ).
% someI2_ex
thf(fact_88_someI__ex,axiom,
! [P: v > $o] :
( ? [X_1: v] : ( P @ X_1 )
=> ( P @ ( fChoice_v @ P ) ) ) ).
% someI_ex
thf(fact_89_someI2,axiom,
! [P: v > $o,A: v,Q: v > $o] :
( ( P @ A )
=> ( ! [X2: v] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( fChoice_v @ P ) ) ) ) ).
% someI2
thf(fact_90_unite__wf__env,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ 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 ) )
=> ( sCC_Bl9196236973127232072t_unit @ successors @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ).
% unite_wf_env
thf(fact_91_unite__sub__env,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ 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 ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ).
% unite_sub_env
thf(fact_92_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 )
=> ( ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( successors @ X2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ X2 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_93_prepostdfs,axiom,
! [Vs: set_v,W: v] :
( ( Vs
= ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) )
=> ( ( Vs != bot_bot_set_v )
=> ( ( W
= ( fChoice_v
@ ^ [X: v] : ( member_v @ X @ Vs ) ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( ( sCC_Bl36166008131615352t_unit @ successors @ W @ ea )
=> ( sCC_Bl8953792750115413617t_unit @ successors @ W @ ea @ ( sCC_Bloemen_dfs_v @ successors @ W @ ea ) ) ) ) ) ) ) ) ).
% prepostdfs
thf(fact_94_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_95_succ__re,axiom,
! [Y4: v,X3: v,Z2: v] :
( ( member_v @ Y4 @ ( successors @ X3 ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y4 @ Z2 )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X3 @ Z2 ) ) ) ).
% succ_re
thf(fact_96_reachable__end_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
= ( ? [X: v] :
( ( A1 = X )
& ( A22 = X ) )
| ? [X: v,Y: v,Z3: v] :
( ( A1 = X )
& ( A22 = Z3 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
& ( member_v @ Z3 @ ( successors @ Y ) ) ) ) ) ).
% reachable_end.simps
thf(fact_97_re__succ,axiom,
! [X3: v,Y4: v,Z2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X3 @ Y4 )
=> ( ( member_v @ Z2 @ ( successors @ Y4 ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X3 @ Z2 ) ) ) ).
% re_succ
thf(fact_98_re__refl,axiom,
! [X3: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X3 @ X3 ) ).
% re_refl
thf(fact_99_reachable__end_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y3 )
=> ~ ( member_v @ A22 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.cases
thf(fact_100_reachable_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( member_v @ Y3 @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ A22 ) ) ) ) ).
% reachable.cases
thf(fact_101_reachable__refl,axiom,
! [X3: v] : ( sCC_Bl649662514949026229able_v @ successors @ X3 @ X3 ) ).
% reachable_refl
thf(fact_102_reachable__succ,axiom,
! [Y4: v,X3: v,Z2: v] :
( ( member_v @ Y4 @ ( successors @ X3 ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y4 @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Z2 ) ) ) ).
% reachable_succ
thf(fact_103_reachable_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
= ( ? [X: v] :
( ( A1 = X )
& ( A22 = X ) )
| ? [X: v,Y: v,Z3: v] :
( ( A1 = X )
& ( A22 = Z3 )
& ( member_v @ Y @ ( successors @ X ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z3 ) ) ) ) ).
% reachable.simps
thf(fact_104_reachable__edge,axiom,
! [Y4: v,X3: v] :
( ( member_v @ Y4 @ ( successors @ X3 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y4 ) ) ).
% reachable_edge
thf(fact_105_reachable__end__induct,axiom,
! [X3: v,Y4: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y4 )
=> ( ! [X2: v] : ( P @ X2 @ X2 )
=> ( ! [X2: v,Y3: v,Z4: v] :
( ( P @ X2 @ Y3 )
=> ( ( member_v @ Z4 @ ( successors @ Y3 ) )
=> ( P @ X2 @ Z4 ) ) )
=> ( P @ X3 @ Y4 ) ) ) ) ).
% reachable_end_induct
thf(fact_106_reachable__trans,axiom,
! [X3: v,Y4: v,Z2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y4 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y4 @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Z2 ) ) ) ).
% reachable_trans
thf(fact_107_succ__reachable,axiom,
! [X3: v,Y4: v,Z2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y4 )
=> ( ( member_v @ Z2 @ ( successors @ Y4 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Z2 ) ) ) ).
% succ_reachable
thf(fact_108_re__reachable,axiom,
! [X3: v,Y4: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X3 @ Y4 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y4 ) ) ).
% re_reachable
thf(fact_109_reachable__re,axiom,
! [X3: v,Y4: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y4 )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X3 @ Y4 ) ) ).
% reachable_re
thf(fact_110_init__env__pre__dfs,axiom,
! [V: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ).
% init_env_pre_dfs
thf(fact_111_e_H__def,axiom,
( ( ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( e2 = ea ) )
& ( ~ ( member_v @ w @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( ( ~ ( member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( e2
= ( sCC_Bloemen_dfs_v @ successors @ w @ ea ) ) )
& ( ( member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( e2
= ( sCC_Bloemen_unite_v @ va @ w @ ea ) ) ) ) ) ) ).
% e'_def
thf(fact_112_sccE,axiom,
! [S: set_v,X3: v,Y4: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( member_v @ X3 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y4 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y4 @ X3 )
=> ( member_v @ Y4 @ S ) ) ) ) ) ).
% sccE
thf(fact_113_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_114_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_115_graph_Odfs_Ocong,axiom,
sCC_Bloemen_dfs_v = sCC_Bloemen_dfs_v ).
% graph.dfs.cong
thf(fact_116_graph_Opost__dfs_Ocong,axiom,
sCC_Bl8953792750115413617t_unit = sCC_Bl8953792750115413617t_unit ).
% graph.post_dfs.cong
thf(fact_117_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_118_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_119_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_120_bot__set__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v @ bot_bot_set_v_o ) ) ).
% bot_set_def
thf(fact_121_is__subscc__def,axiom,
! [S: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
= ( ! [X: v] :
( ( member_v @ X @ S )
=> ! [Y: v] :
( ( member_v @ Y @ S )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ) ) ) ).
% is_subscc_def
thf(fact_122_graph_Ounite__sub__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ 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 ) )
=> ( sCC_Bl7963838319573962697t_unit @ E @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_123_graph_Ounite__sub__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ 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 ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_124_ra__empty,axiom,
! [X3: v,Y4: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y4 @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y4 ) ) ).
% ra_empty
thf(fact_125_ra__reachable,axiom,
! [X3: v,Y4: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y4 @ E4 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y4 ) ) ).
% ra_reachable
thf(fact_126_prepostdfss,axiom,
! [Vs: set_v,W: v,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( Vs
= ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) )
=> ( ( Vs != bot_bot_set_v )
=> ( ( W
= ( fChoice_v
@ ^ [X: v] : ( member_v @ X @ Vs ) ) )
=> ( ( ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( E2 = ea ) )
& ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( E2
= ( sCC_Bloemen_dfs_v @ successors @ W @ ea ) ) )
& ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( E2
= ( sCC_Bloemen_unite_v @ va @ W @ ea ) ) ) ) ) )
=> ( ( E3
= ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X: v] : ( if_set_v @ ( X = va ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ E2 @ va ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ X ) )
@ E2 ) )
=> ( ( sCC_Bl1748261141445803503t_unit @ successors @ va @ E3 )
=> ( sCC_Bl6082031138996704384t_unit @ successors @ va @ E3 @ ( sCC_Bloemen_dfss_v @ successors @ va @ E3 ) ) ) ) ) ) ) ) ).
% prepostdfss
thf(fact_127_graph_Ounite__wf__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ 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 ) )
=> ( sCC_Bl7798947040364291444t_unit @ Successors @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_128_graph_Ounite__wf__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ 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 ) )
=> ( sCC_Bl9196236973127232072t_unit @ Successors @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_129_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_130_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 )
=> ( ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( Successors @ X2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ X2 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_131_verit__sko__ex_H,axiom,
! [P: v > $o,A2: $o] :
( ( ( P @ ( fChoice_v @ P ) )
= A2 )
=> ( ( ? [X5: v] : ( P @ X5 ) )
= A2 ) ) ).
% verit_sko_ex'
thf(fact_132_verit__sko__forall,axiom,
( ( ^ [P3: v > $o] :
! [X6: v] : ( P3 @ X6 ) )
= ( ^ [P4: v > $o] :
( P4
@ ( fChoice_v
@ ^ [X: v] :
~ ( P4 @ X ) ) ) ) ) ).
% verit_sko_forall
thf(fact_133_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_134_ra__trans,axiom,
! [X3: v,Y4: v,E4: set_Product_prod_v_v,Z2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y4 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y4 @ Z2 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Z2 @ E4 ) ) ) ).
% ra_trans
thf(fact_135_ra__refl,axiom,
! [X3: v,E4: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ X3 @ E4 ) ).
% ra_refl
thf(fact_136_insert__absorb2,axiom,
! [X3: v,A2: set_v] :
( ( insert_v @ X3 @ ( insert_v @ X3 @ A2 ) )
= ( insert_v @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_137_insert__absorb2,axiom,
! [X3: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X3 @ ( insert1338601472111419319od_v_v @ X3 @ A2 ) )
= ( insert1338601472111419319od_v_v @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_138_insert__absorb2,axiom,
! [X3: set_v,A2: set_set_v] :
( ( insert_set_v @ X3 @ ( insert_set_v @ X3 @ A2 ) )
= ( insert_set_v @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_139_insert__iff,axiom,
! [A: set_v,B3: set_v,A2: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v @ B3 @ A2 ) )
= ( ( A = B3 )
| ( member_set_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_140_insert__iff,axiom,
! [A: v,B3: v,A2: set_v] :
( ( member_v @ A @ ( insert_v @ B3 @ A2 ) )
= ( ( A = B3 )
| ( member_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_141_insert__iff,axiom,
! [A: product_prod_v_v,B3: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B3 @ A2 ) )
= ( ( A = B3 )
| ( member7453568604450474000od_v_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_142_insertCI,axiom,
! [A: set_v,B: set_set_v,B3: set_v] :
( ( ~ ( member_set_v @ A @ B )
=> ( A = B3 ) )
=> ( member_set_v @ A @ ( insert_set_v @ B3 @ B ) ) ) ).
% insertCI
thf(fact_143_insertCI,axiom,
! [A: v,B: set_v,B3: v] :
( ( ~ ( member_v @ A @ B )
=> ( A = B3 ) )
=> ( member_v @ A @ ( insert_v @ B3 @ B ) ) ) ).
% insertCI
thf(fact_144_insertCI,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B3: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B )
=> ( A = B3 ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B3 @ B ) ) ) ).
% insertCI
thf(fact_145_sup_Oright__idem,axiom,
! [A: set_v,B3: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B3 ) @ B3 )
= ( sup_sup_set_v @ A @ B3 ) ) ).
% sup.right_idem
thf(fact_146_sup_Oright__idem,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B3 ) @ B3 )
= ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ).
% sup.right_idem
thf(fact_147_sup_Oright__idem,axiom,
! [A: set_set_v,B3: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A @ B3 ) @ B3 )
= ( sup_sup_set_set_v @ A @ B3 ) ) ).
% sup.right_idem
thf(fact_148_sup_Oright__idem,axiom,
! [A: product_unit,B3: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ A @ B3 ) @ B3 )
= ( sup_sup_Product_unit @ A @ B3 ) ) ).
% sup.right_idem
thf(fact_149_sup__left__idem,axiom,
! [X3: set_v,Y4: set_v] :
( ( sup_sup_set_v @ X3 @ ( sup_sup_set_v @ X3 @ Y4 ) )
= ( sup_sup_set_v @ X3 @ Y4 ) ) ).
% sup_left_idem
thf(fact_150_sup__left__idem,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) )
= ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) ) ).
% sup_left_idem
thf(fact_151_sup__left__idem,axiom,
! [X3: set_set_v,Y4: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ ( sup_sup_set_set_v @ X3 @ Y4 ) )
= ( sup_sup_set_set_v @ X3 @ Y4 ) ) ).
% sup_left_idem
thf(fact_152_sup__left__idem,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( sup_sup_Product_unit @ X3 @ Y4 ) )
= ( sup_sup_Product_unit @ X3 @ Y4 ) ) ).
% sup_left_idem
thf(fact_153_sup_Oleft__idem,axiom,
! [A: set_v,B3: set_v] :
( ( sup_sup_set_v @ A @ ( sup_sup_set_v @ A @ B3 ) )
= ( sup_sup_set_v @ A @ B3 ) ) ).
% sup.left_idem
thf(fact_154_sup_Oleft__idem,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B3 ) )
= ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ).
% sup.left_idem
thf(fact_155_sup_Oleft__idem,axiom,
! [A: set_set_v,B3: set_set_v] :
( ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ A @ B3 ) )
= ( sup_sup_set_set_v @ A @ B3 ) ) ).
% sup.left_idem
thf(fact_156_sup_Oleft__idem,axiom,
! [A: product_unit,B3: product_unit] :
( ( sup_sup_Product_unit @ A @ ( sup_sup_Product_unit @ A @ B3 ) )
= ( sup_sup_Product_unit @ A @ B3 ) ) ).
% sup.left_idem
thf(fact_157_sup__idem,axiom,
! [X3: set_v] :
( ( sup_sup_set_v @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_158_sup__idem,axiom,
! [X3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_159_sup__idem,axiom,
! [X3: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_160_sup__idem,axiom,
! [X3: product_unit] :
( ( sup_sup_Product_unit @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_161_sup_Oidem,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_162_sup_Oidem,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_163_sup_Oidem,axiom,
! [A: set_set_v] :
( ( sup_sup_set_set_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_164_sup_Oidem,axiom,
! [A: product_unit] :
( ( sup_sup_Product_unit @ A @ A )
= A ) ).
% sup.idem
thf(fact_165_Un__iff,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A2 @ B ) )
= ( ( member_v @ C @ A2 )
| ( member_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_166_Un__iff,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A2 )
| ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_167_Un__iff,axiom,
! [C: set_v,A2: set_set_v,B: set_set_v] :
( ( member_set_v @ C @ ( sup_sup_set_set_v @ A2 @ B ) )
= ( ( member_set_v @ C @ A2 )
| ( member_set_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_168_UnCI,axiom,
! [C: v,B: set_v,A2: set_v] :
( ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ A2 ) )
=> ( member_v @ C @ ( sup_sup_set_v @ A2 @ B ) ) ) ).
% UnCI
thf(fact_169_UnCI,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ A2 ) )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ) ).
% UnCI
thf(fact_170_UnCI,axiom,
! [C: set_v,B: set_set_v,A2: set_set_v] :
( ( ~ ( member_set_v @ C @ B )
=> ( member_set_v @ C @ A2 ) )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A2 @ B ) ) ) ).
% UnCI
thf(fact_171_subscc__add,axiom,
! [S: set_v,X3: v,Y4: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
=> ( ( member_v @ X3 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y4 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y4 @ X3 )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v @ Y4 @ S ) ) ) ) ) ) ).
% subscc_add
thf(fact_172_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_173_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_174_sup__bot__left,axiom,
! [X3: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_175_sup__bot__left,axiom,
! [X3: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_176_sup__bot__left,axiom,
! [X3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_177_sup__bot__left,axiom,
! [X3: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_178_sup__bot__right,axiom,
! [X3: product_unit] :
( ( sup_sup_Product_unit @ X3 @ bot_bot_Product_unit )
= X3 ) ).
% sup_bot_right
thf(fact_179_sup__bot__right,axiom,
! [X3: set_v] :
( ( sup_sup_set_v @ X3 @ bot_bot_set_v )
= X3 ) ).
% sup_bot_right
thf(fact_180_sup__bot__right,axiom,
! [X3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ bot_bo723834152578015283od_v_v )
= X3 ) ).
% sup_bot_right
thf(fact_181_sup__bot__right,axiom,
! [X3: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ bot_bot_set_set_v )
= X3 ) ).
% sup_bot_right
thf(fact_182_bot__eq__sup__iff,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ X3 @ Y4 ) )
= ( ( X3 = bot_bot_Product_unit )
& ( Y4 = bot_bot_Product_unit ) ) ) ).
% bot_eq_sup_iff
thf(fact_183_bot__eq__sup__iff,axiom,
! [X3: set_v,Y4: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ X3 @ Y4 ) )
= ( ( X3 = bot_bot_set_v )
& ( Y4 = bot_bot_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_184_bot__eq__sup__iff,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) )
= ( ( X3 = bot_bo723834152578015283od_v_v )
& ( Y4 = bot_bo723834152578015283od_v_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_185_bot__eq__sup__iff,axiom,
! [X3: set_set_v,Y4: set_set_v] :
( ( bot_bot_set_set_v
= ( sup_sup_set_set_v @ X3 @ Y4 ) )
= ( ( X3 = bot_bot_set_set_v )
& ( Y4 = bot_bot_set_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_186_sup__eq__bot__iff,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( ( sup_sup_Product_unit @ X3 @ Y4 )
= bot_bot_Product_unit )
= ( ( X3 = bot_bot_Product_unit )
& ( Y4 = bot_bot_Product_unit ) ) ) ).
% sup_eq_bot_iff
thf(fact_187_sup__eq__bot__iff,axiom,
! [X3: set_v,Y4: set_v] :
( ( ( sup_sup_set_v @ X3 @ Y4 )
= bot_bot_set_v )
= ( ( X3 = bot_bot_set_v )
& ( Y4 = bot_bot_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_188_sup__eq__bot__iff,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ X3 @ Y4 )
= bot_bo723834152578015283od_v_v )
= ( ( X3 = bot_bo723834152578015283od_v_v )
& ( Y4 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_189_sup__eq__bot__iff,axiom,
! [X3: set_set_v,Y4: set_set_v] :
( ( ( sup_sup_set_set_v @ X3 @ Y4 )
= bot_bot_set_set_v )
= ( ( X3 = bot_bot_set_set_v )
& ( Y4 = bot_bot_set_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_190_sup__bot_Oeq__neutr__iff,axiom,
! [A: product_unit,B3: product_unit] :
( ( ( sup_sup_Product_unit @ A @ B3 )
= bot_bot_Product_unit )
= ( ( A = bot_bot_Product_unit )
& ( B3 = bot_bot_Product_unit ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_191_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_v,B3: set_v] :
( ( ( sup_sup_set_v @ A @ B3 )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B3 = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_192_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B3 )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B3 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_193_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_v,B3: set_set_v] :
( ( ( sup_sup_set_set_v @ A @ B3 )
= bot_bot_set_set_v )
= ( ( A = bot_bot_set_set_v )
& ( B3 = bot_bot_set_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_194_sup__bot_Oleft__neutral,axiom,
! [A: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_195_sup__bot_Oleft__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_196_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_197_sup__bot_Oleft__neutral,axiom,
! [A: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_198_sup__bot_Oneutr__eq__iff,axiom,
! [A: product_unit,B3: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ A @ B3 ) )
= ( ( A = bot_bot_Product_unit )
& ( B3 = bot_bot_Product_unit ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_199_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_v,B3: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A @ B3 ) )
= ( ( A = bot_bot_set_v )
& ( B3 = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_200_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A @ B3 ) )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B3 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_201_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_v,B3: set_set_v] :
( ( bot_bot_set_set_v
= ( sup_sup_set_set_v @ A @ B3 ) )
= ( ( A = bot_bot_set_set_v )
& ( B3 = bot_bot_set_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_202_sup__bot_Oright__neutral,axiom,
! [A: product_unit] :
( ( sup_sup_Product_unit @ A @ bot_bot_Product_unit )
= A ) ).
% sup_bot.right_neutral
thf(fact_203_sup__bot_Oright__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_204_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_205_sup__bot_Oright__neutral,axiom,
! [A: set_set_v] :
( ( sup_sup_set_set_v @ A @ bot_bot_set_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_206_singletonI,axiom,
! [A: v] : ( member_v @ A @ ( insert_v @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_207_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_208_singletonI,axiom,
! [A: set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singletonI
thf(fact_209_Un__empty,axiom,
! [A2: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A2 @ B )
= bot_bot_set_v )
= ( ( A2 = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_210_Un__empty,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_211_Un__empty,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ( sup_sup_set_set_v @ A2 @ B )
= bot_bot_set_set_v )
= ( ( A2 = bot_bot_set_set_v )
& ( B = bot_bot_set_set_v ) ) ) ).
% Un_empty
thf(fact_212_Un__insert__right,axiom,
! [A2: set_v,A: v,B: set_v] :
( ( sup_sup_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( sup_sup_set_v @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_213_Un__insert__right,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_214_Un__insert__right,axiom,
! [A2: set_set_v,A: set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( sup_sup_set_set_v @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_215_Un__insert__left,axiom,
! [A: v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( insert_v @ A @ B ) @ C2 )
= ( insert_v @ A @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_216_Un__insert__left,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_217_Un__insert__left,axiom,
! [A: set_v,B: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( insert_set_v @ A @ B ) @ C2 )
= ( insert_set_v @ A @ ( sup_sup_set_set_v @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_218_insert__Diff1,axiom,
! [X3: set_v,B: set_set_v,A2: set_set_v] :
( ( member_set_v @ X3 @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X3 @ A2 ) @ B )
= ( minus_7228012346218142266_set_v @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_219_insert__Diff1,axiom,
! [X3: product_prod_v_v,B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X3 @ A2 ) @ B )
= ( minus_4183494784930505774od_v_v @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_220_insert__Diff1,axiom,
! [X3: v,B: set_v,A2: set_v] :
( ( member_v @ X3 @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X3 @ A2 ) @ B )
= ( minus_minus_set_v @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_221_Diff__insert0,axiom,
! [X3: set_v,A2: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X3 @ A2 )
=> ( ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ X3 @ B ) )
= ( minus_7228012346218142266_set_v @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_222_Diff__insert0,axiom,
! [X3: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X3 @ B ) )
= ( minus_4183494784930505774od_v_v @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_223_Diff__insert0,axiom,
! [X3: v,A2: set_v,B: set_v] :
( ~ ( member_v @ X3 @ A2 )
=> ( ( minus_minus_set_v @ A2 @ ( insert_v @ X3 @ B ) )
= ( minus_minus_set_v @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_224_Un__Diff__cancel2,axiom,
! [B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B @ A2 ) @ A2 )
= ( sup_su414716646722978715od_v_v @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_225_Un__Diff__cancel2,axiom,
! [B: set_set_v,A2: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ B @ A2 ) @ A2 )
= ( sup_sup_set_set_v @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_226_Un__Diff__cancel2,axiom,
! [B: set_v,A2: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ B @ A2 ) @ A2 )
= ( sup_sup_set_v @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_227_Un__Diff__cancel,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B @ A2 ) )
= ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_228_Un__Diff__cancel,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B @ A2 ) )
= ( sup_sup_set_set_v @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_229_Un__Diff__cancel,axiom,
! [A2: set_v,B: set_v] :
( ( sup_sup_set_v @ A2 @ ( minus_minus_set_v @ B @ A2 ) )
= ( sup_sup_set_v @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_230_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_231_singleton__conv,axiom,
! [A: v] :
( ( collect_v
@ ^ [X: v] : ( X = A ) )
= ( insert_v @ A @ bot_bot_set_v ) ) ).
% singleton_conv
thf(fact_232_singleton__conv,axiom,
! [A: product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] : ( X = A ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singleton_conv
thf(fact_233_singleton__conv,axiom,
! [A: set_v] :
( ( collect_set_v
@ ^ [X: set_v] : ( X = A ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singleton_conv
thf(fact_234_singleton__conv2,axiom,
! [A: v] :
( ( collect_v
@ ( ^ [Y2: v,Z: v] : ( Y2 = Z )
@ A ) )
= ( insert_v @ A @ bot_bot_set_v ) ) ).
% singleton_conv2
thf(fact_235_singleton__conv2,axiom,
! [A: product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ( ^ [Y2: product_prod_v_v,Z: product_prod_v_v] : ( Y2 = Z )
@ A ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singleton_conv2
thf(fact_236_singleton__conv2,axiom,
! [A: set_v] :
( ( collect_set_v
@ ( ^ [Y2: set_v,Z: set_v] : ( Y2 = Z )
@ A ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singleton_conv2
thf(fact_237_pre__dfss__explored__pre__dfss,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X: v] : ( if_set_v @ ( X = V ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X ) )
@ E ) ) ) ) ) ).
% pre_dfss_explored_pre_dfss
thf(fact_238_insert__Diff__single,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_239_insert__Diff__single,axiom,
! [A: set_v,A2: set_set_v] :
( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
= ( insert_set_v @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_240_insert__Diff__single,axiom,
! [A: v,A2: set_v] :
( ( insert_v @ A @ ( minus_minus_set_v @ A2 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= ( insert_v @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_241_pre__dfss__unite__pre__dfss,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ 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 ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X: v] : ( if_set_v @ ( X = V ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ V ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ X ) )
@ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% pre_dfss_unite_pre_dfss
thf(fact_242_pre__dfss__post__dfs__pre__dfss,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ W @ E @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X: v] : ( if_set_v @ ( X = V ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ V ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ X ) )
@ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) ) ) ) ) ) ) ).
% pre_dfss_post_dfs_pre_dfss
thf(fact_243_e_H_H__def,axiom,
( e
= ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X: v] : ( if_set_v @ ( X = va ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ e2 @ va ) @ ( insert_v @ w @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ e2 @ X ) )
@ e2 ) ) ).
% e''_def
thf(fact_244_graph_Osubscc__add,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S: set_set_v,X3: set_v,Y4: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7907073126578335045_set_v @ Successors @ S )
=> ( ( member_set_v @ X3 @ S )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ X3 @ Y4 )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ Y4 @ X3 )
=> ( sCC_Bl7907073126578335045_set_v @ Successors @ ( insert_set_v @ Y4 @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_245_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,X3: product_prod_v_v,Y4: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X3 @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X3 @ Y4 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y4 @ X3 )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y4 @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_246_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X3: v,Y4: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
=> ( ( member_v @ X3 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y4 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y4 @ X3 )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v @ Y4 @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_247_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v,E4: set_Product_prod_v_v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y4 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y4 @ Z2 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Z2 @ E4 ) ) ) ) ).
% graph.ra_trans
thf(fact_248_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_249_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ X3 @ E4 ) ) ).
% graph.ra_refl
thf(fact_250_mk__disjoint__insert,axiom,
! [A: set_v,A2: set_set_v] :
( ( member_set_v @ A @ A2 )
=> ? [B4: set_set_v] :
( ( A2
= ( insert_set_v @ A @ B4 ) )
& ~ ( member_set_v @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_251_mk__disjoint__insert,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ? [B4: set_v] :
( ( A2
= ( insert_v @ A @ B4 ) )
& ~ ( member_v @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_252_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ? [B4: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ A @ B4 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_253_Un__left__commute,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ B @ ( sup_sup_set_v @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_254_Un__left__commute,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_255_Un__left__commute,axiom,
! [A2: set_set_v,B: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( sup_sup_set_set_v @ B @ C2 ) )
= ( sup_sup_set_set_v @ B @ ( sup_sup_set_set_v @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_256_insert__commute,axiom,
! [X3: v,Y4: v,A2: set_v] :
( ( insert_v @ X3 @ ( insert_v @ Y4 @ A2 ) )
= ( insert_v @ Y4 @ ( insert_v @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_257_insert__commute,axiom,
! [X3: product_prod_v_v,Y4: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X3 @ ( insert1338601472111419319od_v_v @ Y4 @ A2 ) )
= ( insert1338601472111419319od_v_v @ Y4 @ ( insert1338601472111419319od_v_v @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_258_insert__commute,axiom,
! [X3: set_v,Y4: set_v,A2: set_set_v] :
( ( insert_set_v @ X3 @ ( insert_set_v @ Y4 @ A2 ) )
= ( insert_set_v @ Y4 @ ( insert_set_v @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_259_Un__left__absorb,axiom,
! [A2: set_v,B: set_v] :
( ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ A2 @ B ) )
= ( sup_sup_set_v @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_260_Un__left__absorb,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ A2 @ B ) )
= ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_261_Un__left__absorb,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( sup_sup_set_set_v @ A2 @ B ) )
= ( sup_sup_set_set_v @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_262_insert__eq__iff,axiom,
! [A: set_v,A2: set_set_v,B3: set_v,B: set_set_v] :
( ~ ( member_set_v @ A @ A2 )
=> ( ~ ( member_set_v @ B3 @ B )
=> ( ( ( insert_set_v @ A @ A2 )
= ( insert_set_v @ B3 @ B ) )
= ( ( ( A = B3 )
=> ( A2 = B ) )
& ( ( A != B3 )
=> ? [C3: set_set_v] :
( ( A2
= ( insert_set_v @ B3 @ C3 ) )
& ~ ( member_set_v @ B3 @ C3 )
& ( B
= ( insert_set_v @ A @ C3 ) )
& ~ ( member_set_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_263_insert__eq__iff,axiom,
! [A: v,A2: set_v,B3: v,B: set_v] :
( ~ ( member_v @ A @ A2 )
=> ( ~ ( member_v @ B3 @ B )
=> ( ( ( insert_v @ A @ A2 )
= ( insert_v @ B3 @ B ) )
= ( ( ( A = B3 )
=> ( A2 = B ) )
& ( ( A != B3 )
=> ? [C3: set_v] :
( ( A2
= ( insert_v @ B3 @ C3 ) )
& ~ ( member_v @ B3 @ C3 )
& ( B
= ( insert_v @ A @ C3 ) )
& ~ ( member_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_264_insert__eq__iff,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B3: product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ B3 @ B )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A2 )
= ( insert1338601472111419319od_v_v @ B3 @ B ) )
= ( ( ( A = B3 )
=> ( A2 = B ) )
& ( ( A != B3 )
=> ? [C3: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ B3 @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B3 @ C3 )
& ( B
= ( insert1338601472111419319od_v_v @ A @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_265_insert__absorb,axiom,
! [A: set_v,A2: set_set_v] :
( ( member_set_v @ A @ A2 )
=> ( ( insert_set_v @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_266_insert__absorb,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( insert_v @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_267_insert__absorb,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( insert1338601472111419319od_v_v @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_268_insert__ident,axiom,
! [X3: set_v,A2: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X3 @ A2 )
=> ( ~ ( member_set_v @ X3 @ B )
=> ( ( ( insert_set_v @ X3 @ A2 )
= ( insert_set_v @ X3 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_269_insert__ident,axiom,
! [X3: v,A2: set_v,B: set_v] :
( ~ ( member_v @ X3 @ A2 )
=> ( ~ ( member_v @ X3 @ B )
=> ( ( ( insert_v @ X3 @ A2 )
= ( insert_v @ X3 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_270_insert__ident,axiom,
! [X3: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ B )
=> ( ( ( insert1338601472111419319od_v_v @ X3 @ A2 )
= ( insert1338601472111419319od_v_v @ X3 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_271_Set_Oset__insert,axiom,
! [X3: set_v,A2: set_set_v] :
( ( member_set_v @ X3 @ A2 )
=> ~ ! [B4: set_set_v] :
( ( A2
= ( insert_set_v @ X3 @ B4 ) )
=> ( member_set_v @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_272_Set_Oset__insert,axiom,
! [X3: v,A2: set_v] :
( ( member_v @ X3 @ A2 )
=> ~ ! [B4: set_v] :
( ( A2
= ( insert_v @ X3 @ B4 ) )
=> ( member_v @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_273_Set_Oset__insert,axiom,
! [X3: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ~ ! [B4: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ X3 @ B4 ) )
=> ( member7453568604450474000od_v_v @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_274_insert__def,axiom,
( insert_v
= ( ^ [A4: v] :
( sup_sup_set_v
@ ( collect_v
@ ^ [X: v] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_275_insert__def,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A4: product_prod_v_v] :
( sup_su414716646722978715od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_276_insert__def,axiom,
( insert_set_v
= ( ^ [A4: set_v] :
( sup_sup_set_set_v
@ ( collect_set_v
@ ^ [X: set_v] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_277_Un__commute,axiom,
( sup_sup_set_v
= ( ^ [A3: set_v,B2: set_v] : ( sup_sup_set_v @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_278_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_279_Un__commute,axiom,
( sup_sup_set_set_v
= ( ^ [A3: set_set_v,B2: set_set_v] : ( sup_sup_set_set_v @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_280_Un__absorb,axiom,
! [A2: set_v] :
( ( sup_sup_set_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_281_Un__absorb,axiom,
! [A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_282_Un__absorb,axiom,
! [A2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_283_insertI2,axiom,
! [A: set_v,B: set_set_v,B3: set_v] :
( ( member_set_v @ A @ B )
=> ( member_set_v @ A @ ( insert_set_v @ B3 @ B ) ) ) ).
% insertI2
thf(fact_284_insertI2,axiom,
! [A: v,B: set_v,B3: v] :
( ( member_v @ A @ B )
=> ( member_v @ A @ ( insert_v @ B3 @ B ) ) ) ).
% insertI2
thf(fact_285_insertI2,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B3 @ B ) ) ) ).
% insertI2
thf(fact_286_insertI1,axiom,
! [A: set_v,B: set_set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ B ) ) ).
% insertI1
thf(fact_287_insertI1,axiom,
! [A: v,B: set_v] : ( member_v @ A @ ( insert_v @ A @ B ) ) ).
% insertI1
thf(fact_288_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_289_Un__assoc,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A2 @ B ) @ C2 )
= ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_290_Un__assoc,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ C2 )
= ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_291_Un__assoc,axiom,
! [A2: set_set_v,B: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ C2 )
= ( sup_sup_set_set_v @ A2 @ ( sup_sup_set_set_v @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_292_insertE,axiom,
! [A: set_v,B3: set_v,A2: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v @ B3 @ A2 ) )
=> ( ( A != B3 )
=> ( member_set_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_293_insertE,axiom,
! [A: v,B3: v,A2: set_v] :
( ( member_v @ A @ ( insert_v @ B3 @ A2 ) )
=> ( ( A != B3 )
=> ( member_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_294_insertE,axiom,
! [A: product_prod_v_v,B3: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B3 @ A2 ) )
=> ( ( A != B3 )
=> ( member7453568604450474000od_v_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_295_ball__Un,axiom,
! [A2: set_v,B: set_v,P: v > $o] :
( ( ! [X: v] :
( ( member_v @ X @ ( sup_sup_set_v @ A2 @ B ) )
=> ( P @ X ) ) )
= ( ! [X: v] :
( ( member_v @ X @ A2 )
=> ( P @ X ) )
& ! [X: v] :
( ( member_v @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_296_ball__Un,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A2 @ B ) )
=> ( P @ X ) ) )
= ( ! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A2 )
=> ( P @ X ) )
& ! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_297_ball__Un,axiom,
! [A2: set_set_v,B: set_set_v,P: set_v > $o] :
( ( ! [X: set_v] :
( ( member_set_v @ X @ ( sup_sup_set_set_v @ A2 @ B ) )
=> ( P @ X ) ) )
= ( ! [X: set_v] :
( ( member_set_v @ X @ A2 )
=> ( P @ X ) )
& ! [X: set_v] :
( ( member_set_v @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_298_bex__Un,axiom,
! [A2: set_v,B: set_v,P: v > $o] :
( ( ? [X: v] :
( ( member_v @ X @ ( sup_sup_set_v @ A2 @ B ) )
& ( P @ X ) ) )
= ( ? [X: v] :
( ( member_v @ X @ A2 )
& ( P @ X ) )
| ? [X: v] :
( ( member_v @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_299_bex__Un,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ? [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A2 @ B ) )
& ( P @ X ) ) )
= ( ? [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A2 )
& ( P @ X ) )
| ? [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_300_bex__Un,axiom,
! [A2: set_set_v,B: set_set_v,P: set_v > $o] :
( ( ? [X: set_v] :
( ( member_set_v @ X @ ( sup_sup_set_set_v @ A2 @ B ) )
& ( P @ X ) ) )
= ( ? [X: set_v] :
( ( member_set_v @ X @ A2 )
& ( P @ X ) )
| ? [X: set_v] :
( ( member_set_v @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_301_UnI2,axiom,
! [C: v,B: set_v,A2: set_v] :
( ( member_v @ C @ B )
=> ( member_v @ C @ ( sup_sup_set_v @ A2 @ B ) ) ) ).
% UnI2
thf(fact_302_UnI2,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ) ).
% UnI2
thf(fact_303_UnI2,axiom,
! [C: set_v,B: set_set_v,A2: set_set_v] :
( ( member_set_v @ C @ B )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A2 @ B ) ) ) ).
% UnI2
thf(fact_304_UnI1,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ A2 )
=> ( member_v @ C @ ( sup_sup_set_v @ A2 @ B ) ) ) ).
% UnI1
thf(fact_305_UnI1,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ) ).
% UnI1
thf(fact_306_UnI1,axiom,
! [C: set_v,A2: set_set_v,B: set_set_v] :
( ( member_set_v @ C @ A2 )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A2 @ B ) ) ) ).
% UnI1
thf(fact_307_UnE,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A2 @ B ) )
=> ( ~ ( member_v @ C @ A2 )
=> ( member_v @ C @ B ) ) ) ).
% UnE
thf(fact_308_UnE,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B ) )
=> ( ~ ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% UnE
thf(fact_309_UnE,axiom,
! [C: set_v,A2: set_set_v,B: set_set_v] :
( ( member_set_v @ C @ ( sup_sup_set_set_v @ A2 @ B ) )
=> ( ~ ( member_set_v @ C @ A2 )
=> ( member_set_v @ C @ B ) ) ) ).
% UnE
thf(fact_310_sup__left__commute,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( sup_sup_set_v @ X3 @ ( sup_sup_set_v @ Y4 @ Z2 ) )
= ( sup_sup_set_v @ Y4 @ ( sup_sup_set_v @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_311_sup__left__commute,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y4 @ Z2 ) )
= ( sup_su414716646722978715od_v_v @ Y4 @ ( sup_su414716646722978715od_v_v @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_312_sup__left__commute,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ ( sup_sup_set_set_v @ Y4 @ Z2 ) )
= ( sup_sup_set_set_v @ Y4 @ ( sup_sup_set_set_v @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_313_sup__left__commute,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) )
= ( sup_sup_Product_unit @ Y4 @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_314_sup_Oleft__commute,axiom,
! [B3: set_v,A: set_v,C: set_v] :
( ( sup_sup_set_v @ B3 @ ( sup_sup_set_v @ A @ C ) )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B3 @ C ) ) ) ).
% sup.left_commute
thf(fact_315_sup_Oleft__commute,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ B3 @ ( sup_su414716646722978715od_v_v @ A @ C ) )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B3 @ C ) ) ) ).
% sup.left_commute
thf(fact_316_sup_Oleft__commute,axiom,
! [B3: set_set_v,A: set_set_v,C: set_set_v] :
( ( sup_sup_set_set_v @ B3 @ ( sup_sup_set_set_v @ A @ C ) )
= ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ B3 @ C ) ) ) ).
% sup.left_commute
thf(fact_317_sup_Oleft__commute,axiom,
! [B3: product_unit,A: product_unit,C: product_unit] :
( ( sup_sup_Product_unit @ B3 @ ( sup_sup_Product_unit @ A @ C ) )
= ( sup_sup_Product_unit @ A @ ( sup_sup_Product_unit @ B3 @ C ) ) ) ).
% sup.left_commute
thf(fact_318_sup__commute,axiom,
( sup_sup_set_v
= ( ^ [X: set_v,Y: set_v] : ( sup_sup_set_v @ Y @ X ) ) ) ).
% sup_commute
thf(fact_319_sup__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y @ X ) ) ) ).
% sup_commute
thf(fact_320_sup__commute,axiom,
( sup_sup_set_set_v
= ( ^ [X: set_set_v,Y: set_set_v] : ( sup_sup_set_set_v @ Y @ X ) ) ) ).
% sup_commute
thf(fact_321_sup__commute,axiom,
( sup_sup_Product_unit
= ( ^ [X: product_unit,Y: product_unit] : ( sup_sup_Product_unit @ Y @ X ) ) ) ).
% sup_commute
thf(fact_322_sup_Ocommute,axiom,
( sup_sup_set_v
= ( ^ [A4: set_v,B5: set_v] : ( sup_sup_set_v @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_323_sup_Ocommute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A4: set_Product_prod_v_v,B5: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_324_sup_Ocommute,axiom,
( sup_sup_set_set_v
= ( ^ [A4: set_set_v,B5: set_set_v] : ( sup_sup_set_set_v @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_325_sup_Ocommute,axiom,
( sup_sup_Product_unit
= ( ^ [A4: product_unit,B5: product_unit] : ( sup_sup_Product_unit @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_326_sup__assoc,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X3 @ Y4 ) @ Z2 )
= ( sup_sup_set_v @ X3 @ ( sup_sup_set_v @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_327_sup__assoc,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) @ Z2 )
= ( sup_su414716646722978715od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_328_sup__assoc,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ X3 @ Y4 ) @ Z2 )
= ( sup_sup_set_set_v @ X3 @ ( sup_sup_set_set_v @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_329_sup__assoc,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ Z2 )
= ( sup_sup_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_330_sup_Oassoc,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B3 ) @ C )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B3 @ C ) ) ) ).
% sup.assoc
thf(fact_331_sup_Oassoc,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B3 ) @ C )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B3 @ C ) ) ) ).
% sup.assoc
thf(fact_332_sup_Oassoc,axiom,
! [A: set_set_v,B3: set_set_v,C: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A @ B3 ) @ C )
= ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ B3 @ C ) ) ) ).
% sup.assoc
thf(fact_333_sup_Oassoc,axiom,
! [A: product_unit,B3: product_unit,C: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ A @ B3 ) @ C )
= ( sup_sup_Product_unit @ A @ ( sup_sup_Product_unit @ B3 @ C ) ) ) ).
% sup.assoc
thf(fact_334_inf__sup__aci_I5_J,axiom,
( sup_sup_set_v
= ( ^ [X: set_v,Y: set_v] : ( sup_sup_set_v @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_335_inf__sup__aci_I5_J,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_336_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_v
= ( ^ [X: set_set_v,Y: set_set_v] : ( sup_sup_set_set_v @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_337_inf__sup__aci_I5_J,axiom,
( sup_sup_Product_unit
= ( ^ [X: product_unit,Y: product_unit] : ( sup_sup_Product_unit @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_338_inf__sup__aci_I6_J,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X3 @ Y4 ) @ Z2 )
= ( sup_sup_set_v @ X3 @ ( sup_sup_set_v @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_339_inf__sup__aci_I6_J,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) @ Z2 )
= ( sup_su414716646722978715od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_340_inf__sup__aci_I6_J,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ X3 @ Y4 ) @ Z2 )
= ( sup_sup_set_set_v @ X3 @ ( sup_sup_set_set_v @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_341_inf__sup__aci_I6_J,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ Z2 )
= ( sup_sup_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_342_inf__sup__aci_I7_J,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( sup_sup_set_v @ X3 @ ( sup_sup_set_v @ Y4 @ Z2 ) )
= ( sup_sup_set_v @ Y4 @ ( sup_sup_set_v @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_343_inf__sup__aci_I7_J,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y4 @ Z2 ) )
= ( sup_su414716646722978715od_v_v @ Y4 @ ( sup_su414716646722978715od_v_v @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_344_inf__sup__aci_I7_J,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ ( sup_sup_set_set_v @ Y4 @ Z2 ) )
= ( sup_sup_set_set_v @ Y4 @ ( sup_sup_set_set_v @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_345_inf__sup__aci_I7_J,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) )
= ( sup_sup_Product_unit @ Y4 @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_346_inf__sup__aci_I8_J,axiom,
! [X3: set_v,Y4: set_v] :
( ( sup_sup_set_v @ X3 @ ( sup_sup_set_v @ X3 @ Y4 ) )
= ( sup_sup_set_v @ X3 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_347_inf__sup__aci_I8_J,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) )
= ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_348_inf__sup__aci_I8_J,axiom,
! [X3: set_set_v,Y4: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ ( sup_sup_set_set_v @ X3 @ Y4 ) )
= ( sup_sup_set_set_v @ X3 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_349_inf__sup__aci_I8_J,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( sup_sup_Product_unit @ X3 @ Y4 ) )
= ( sup_sup_Product_unit @ X3 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_350_insert__is__Un,axiom,
( insert_v
= ( ^ [A4: v] : ( sup_sup_set_v @ ( insert_v @ A4 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_351_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A4: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_352_insert__is__Un,axiom,
( insert_set_v
= ( ^ [A4: set_v] : ( sup_sup_set_set_v @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_353_Un__singleton__iff,axiom,
! [A2: set_v,B: set_v,X3: v] :
( ( ( sup_sup_set_v @ A2 @ B )
= ( insert_v @ X3 @ bot_bot_set_v ) )
= ( ( ( A2 = bot_bot_set_v )
& ( B
= ( insert_v @ X3 @ bot_bot_set_v ) ) )
| ( ( A2
= ( insert_v @ X3 @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A2
= ( insert_v @ X3 @ bot_bot_set_v ) )
& ( B
= ( insert_v @ X3 @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_354_Un__singleton__iff,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,X3: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A2 @ B )
= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_355_Un__singleton__iff,axiom,
! [A2: set_set_v,B: set_set_v,X3: set_v] :
( ( ( sup_sup_set_set_v @ A2 @ B )
= ( insert_set_v @ X3 @ bot_bot_set_set_v ) )
= ( ( ( A2 = bot_bot_set_set_v )
& ( B
= ( insert_set_v @ X3 @ bot_bot_set_set_v ) ) )
| ( ( A2
= ( insert_set_v @ X3 @ bot_bot_set_set_v ) )
& ( B = bot_bot_set_set_v ) )
| ( ( A2
= ( insert_set_v @ X3 @ bot_bot_set_set_v ) )
& ( B
= ( insert_set_v @ X3 @ bot_bot_set_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_356_singleton__Un__iff,axiom,
! [X3: v,A2: set_v,B: set_v] :
( ( ( insert_v @ X3 @ bot_bot_set_v )
= ( sup_sup_set_v @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_v )
& ( B
= ( insert_v @ X3 @ bot_bot_set_v ) ) )
| ( ( A2
= ( insert_v @ X3 @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A2
= ( insert_v @ X3 @ bot_bot_set_v ) )
& ( B
= ( insert_v @ X3 @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_357_singleton__Un__iff,axiom,
! [X3: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A2 @ B ) )
= ( ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_358_singleton__Un__iff,axiom,
! [X3: set_v,A2: set_set_v,B: set_set_v] :
( ( ( insert_set_v @ X3 @ bot_bot_set_set_v )
= ( sup_sup_set_set_v @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_set_v )
& ( B
= ( insert_set_v @ X3 @ bot_bot_set_set_v ) ) )
| ( ( A2
= ( insert_set_v @ X3 @ bot_bot_set_set_v ) )
& ( B = bot_bot_set_set_v ) )
| ( ( A2
= ( insert_set_v @ X3 @ bot_bot_set_set_v ) )
& ( B
= ( insert_set_v @ X3 @ bot_bot_set_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_359_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_360_Un__def,axiom,
( sup_sup_set_v
= ( ^ [A3: set_v,B2: set_v] :
( collect_v
@ ^ [X: v] :
( ( member_v @ X @ A3 )
| ( member_v @ X @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_361_Un__def,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
| ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_362_Un__def,axiom,
( sup_sup_set_set_v
= ( ^ [A3: set_set_v,B2: set_set_v] :
( collect_set_v
@ ^ [X: set_v] :
( ( member_set_v @ X @ A3 )
| ( member_set_v @ X @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_363_Collect__disj__eq,axiom,
! [P: v > $o,Q: v > $o] :
( ( collect_v
@ ^ [X: v] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_364_Collect__disj__eq,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_su414716646722978715od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_365_Collect__disj__eq,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( collect_set_v
@ ^ [X: set_v] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_366_insert__compr,axiom,
( insert_v
= ( ^ [A4: v,B2: set_v] :
( collect_v
@ ^ [X: v] :
( ( X = A4 )
| ( member_v @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_367_insert__compr,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A4: product_prod_v_v,B2: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] :
( ( X = A4 )
| ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_368_insert__compr,axiom,
( insert_set_v
= ( ^ [A4: set_v,B2: set_set_v] :
( collect_set_v
@ ^ [X: set_v] :
( ( X = A4 )
| ( member_set_v @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_369_insert__Collect,axiom,
! [A: v,P: v > $o] :
( ( insert_v @ A @ ( collect_v @ P ) )
= ( collect_v
@ ^ [U: v] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_370_insert__Collect,axiom,
! [A: product_prod_v_v,P: product_prod_v_v > $o] :
( ( insert1338601472111419319od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
= ( collec140062887454715474od_v_v
@ ^ [U: product_prod_v_v] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_371_insert__Collect,axiom,
! [A: set_v,P: set_v > $o] :
( ( insert_set_v @ A @ ( collect_set_v @ P ) )
= ( collect_set_v
@ ^ [U: set_v] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_372_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_373_graph_Ora__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y4 @ E4 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y4 ) ) ) ).
% graph.ra_reachable
thf(fact_374_graph_Ora__empty,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y4 @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y4 ) ) ) ).
% graph.ra_empty
thf(fact_375_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_376_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_377_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 )
= ( ! [X: v] :
( ( member_v @ X @ S )
=> ! [Y: v] :
( ( member_v @ Y @ S )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_378_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,X3: product_prod_v_v,Y4: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X3 @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X3 @ Y4 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y4 @ X3 )
=> ( member7453568604450474000od_v_v @ Y4 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_379_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X3: v,Y4: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( member_v @ X3 @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y4 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y4 @ X3 )
=> ( member_v @ Y4 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_380_graph_Opost__dfss_Ocong,axiom,
sCC_Bl6082031138996704384t_unit = sCC_Bl6082031138996704384t_unit ).
% graph.post_dfss.cong
thf(fact_381_Un__empty__left,axiom,
! [B: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B )
= B ) ).
% Un_empty_left
thf(fact_382_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_383_Un__empty__left,axiom,
! [B: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ B )
= B ) ).
% Un_empty_left
thf(fact_384_Un__empty__right,axiom,
! [A2: set_v] :
( ( sup_sup_set_v @ A2 @ bot_bot_set_v )
= A2 ) ).
% Un_empty_right
thf(fact_385_Un__empty__right,axiom,
! [A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= A2 ) ).
% Un_empty_right
thf(fact_386_Un__empty__right,axiom,
! [A2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ bot_bot_set_set_v )
= A2 ) ).
% Un_empty_right
thf(fact_387_singletonD,axiom,
! [B3: v,A: v] :
( ( member_v @ B3 @ ( insert_v @ A @ bot_bot_set_v ) )
=> ( B3 = A ) ) ).
% singletonD
thf(fact_388_singletonD,axiom,
! [B3: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B3 = A ) ) ).
% singletonD
thf(fact_389_singletonD,axiom,
! [B3: set_v,A: set_v] :
( ( member_set_v @ B3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
=> ( B3 = A ) ) ).
% singletonD
thf(fact_390_singleton__iff,axiom,
! [B3: v,A: v] :
( ( member_v @ B3 @ ( insert_v @ A @ bot_bot_set_v ) )
= ( B3 = A ) ) ).
% singleton_iff
thf(fact_391_singleton__iff,axiom,
! [B3: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B3 = A ) ) ).
% singleton_iff
thf(fact_392_singleton__iff,axiom,
! [B3: set_v,A: set_v] :
( ( member_set_v @ B3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
= ( B3 = A ) ) ).
% singleton_iff
thf(fact_393_doubleton__eq__iff,axiom,
! [A: v,B3: v,C: v,D: v] :
( ( ( insert_v @ A @ ( insert_v @ B3 @ bot_bot_set_v ) )
= ( insert_v @ C @ ( insert_v @ D @ bot_bot_set_v ) ) )
= ( ( ( A = C )
& ( B3 = D ) )
| ( ( A = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_394_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B3: product_prod_v_v,C: product_prod_v_v,D: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C )
& ( B3 = D ) )
| ( ( A = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_395_doubleton__eq__iff,axiom,
! [A: set_v,B3: set_v,C: set_v,D: set_v] :
( ( ( insert_set_v @ A @ ( insert_set_v @ B3 @ bot_bot_set_set_v ) )
= ( insert_set_v @ C @ ( insert_set_v @ D @ bot_bot_set_set_v ) ) )
= ( ( ( A = C )
& ( B3 = D ) )
| ( ( A = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_396_insert__not__empty,axiom,
! [A: v,A2: set_v] :
( ( insert_v @ A @ A2 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_397_insert__not__empty,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ A2 )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_398_insert__not__empty,axiom,
! [A: set_v,A2: set_set_v] :
( ( insert_set_v @ A @ A2 )
!= bot_bot_set_set_v ) ).
% insert_not_empty
thf(fact_399_singleton__inject,axiom,
! [A: v,B3: v] :
( ( ( insert_v @ A @ bot_bot_set_v )
= ( insert_v @ B3 @ bot_bot_set_v ) )
=> ( A = B3 ) ) ).
% singleton_inject
thf(fact_400_singleton__inject,axiom,
! [A: product_prod_v_v,B3: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v ) )
=> ( A = B3 ) ) ).
% singleton_inject
thf(fact_401_singleton__inject,axiom,
! [A: set_v,B3: set_v] :
( ( ( insert_set_v @ A @ bot_bot_set_set_v )
= ( insert_set_v @ B3 @ bot_bot_set_set_v ) )
=> ( A = B3 ) ) ).
% singleton_inject
thf(fact_402_Un__Diff,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ C2 )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ C2 ) @ ( minus_4183494784930505774od_v_v @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_403_Un__Diff,axiom,
! [A2: set_set_v,B: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ C2 )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A2 @ C2 ) @ ( minus_7228012346218142266_set_v @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_404_Un__Diff,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( sup_sup_set_v @ A2 @ B ) @ C2 )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ C2 ) @ ( minus_minus_set_v @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_405_graph_Oreachable__edge,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y4: product_prod_v_v,X3: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y4 @ ( Successors @ X3 ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X3 @ Y4 ) ) ) ).
% graph.reachable_edge
thf(fact_406_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y4: v,X3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y4 @ ( Successors @ X3 ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y4 ) ) ) ).
% graph.reachable_edge
thf(fact_407_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X3: product_prod_v_v,Y4: product_prod_v_v,Z2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X3 @ Y4 )
=> ( ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y4 ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X3 @ Z2 ) ) ) ) ).
% graph.succ_reachable
thf(fact_408_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y4 )
=> ( ( member_v @ Z2 @ ( Successors @ Y4 ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Z2 ) ) ) ) ).
% graph.succ_reachable
thf(fact_409_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,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_410_graph_Oreachable_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( member_v @ Y3 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_411_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,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A22 )
= ( ? [X: product_prod_v_v] :
( ( A1 = X )
& ( A22 = X ) )
| ? [X: product_prod_v_v,Y: product_prod_v_v,Z3: product_prod_v_v] :
( ( A1 = X )
& ( A22 = Z3 )
& ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ Z3 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_412_graph_Oreachable_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A22 )
= ( ? [X: v] :
( ( A1 = X )
& ( A22 = X ) )
| ? [X: v,Y: v,Z3: v] :
( ( A1 = X )
& ( A22 = Z3 )
& ( member_v @ Y @ ( Successors @ X ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z3 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_413_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y4 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y4 @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Z2 ) ) ) ) ).
% graph.reachable_trans
thf(fact_414_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X3: product_prod_v_v,Y4: 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 @ X3 @ Y4 )
=> ( ! [X2: product_prod_v_v] : ( P @ X2 @ X2 )
=> ( ! [X2: product_prod_v_v,Y3: product_prod_v_v,Z4: product_prod_v_v] :
( ( P @ X2 @ Y3 )
=> ( ( member7453568604450474000od_v_v @ Z4 @ ( Successors @ Y3 ) )
=> ( P @ X2 @ Z4 ) ) )
=> ( P @ X3 @ Y4 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_415_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y4 )
=> ( ! [X2: v] : ( P @ X2 @ X2 )
=> ( ! [X2: v,Y3: v,Z4: v] :
( ( P @ X2 @ Y3 )
=> ( ( member_v @ Z4 @ ( Successors @ Y3 ) )
=> ( P @ X2 @ Z4 ) ) )
=> ( P @ X3 @ Y4 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_416_graph_Oreachable__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ X3 ) ) ).
% graph.reachable_refl
thf(fact_417_graph_Oreachable__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y4: product_prod_v_v,X3: product_prod_v_v,Z2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y4 @ ( Successors @ X3 ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y4 @ Z2 )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X3 @ Z2 ) ) ) ) ).
% graph.reachable_succ
thf(fact_418_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y4: v,X3: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y4 @ ( Successors @ X3 ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y4 @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Z2 ) ) ) ) ).
% graph.reachable_succ
thf(fact_419_insert__Diff__if,axiom,
! [X3: set_v,B: set_set_v,A2: set_set_v] :
( ( ( member_set_v @ X3 @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X3 @ A2 ) @ B )
= ( minus_7228012346218142266_set_v @ A2 @ B ) ) )
& ( ~ ( member_set_v @ X3 @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X3 @ A2 ) @ B )
= ( insert_set_v @ X3 @ ( minus_7228012346218142266_set_v @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_420_insert__Diff__if,axiom,
! [X3: product_prod_v_v,B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X3 @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X3 @ A2 ) @ B )
= ( minus_4183494784930505774od_v_v @ A2 @ B ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X3 @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X3 @ A2 ) @ B )
= ( insert1338601472111419319od_v_v @ X3 @ ( minus_4183494784930505774od_v_v @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_421_insert__Diff__if,axiom,
! [X3: v,B: set_v,A2: set_v] :
( ( ( member_v @ X3 @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X3 @ A2 ) @ B )
= ( minus_minus_set_v @ A2 @ B ) ) )
& ( ~ ( member_v @ X3 @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X3 @ A2 ) @ B )
= ( insert_v @ X3 @ ( minus_minus_set_v @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_422_graph_Osucc__re,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y4: product_prod_v_v,X3: product_prod_v_v,Z2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y4 @ ( Successors @ X3 ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y4 @ Z2 )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X3 @ Z2 ) ) ) ) ).
% graph.succ_re
thf(fact_423_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y4: v,X3: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y4 @ ( Successors @ X3 ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y4 @ Z2 )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X3 @ Z2 ) ) ) ) ).
% graph.succ_re
thf(fact_424_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,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y3 )
=> ~ ( member7453568604450474000od_v_v @ A22 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_425_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y3 )
=> ~ ( member_v @ A22 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_426_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,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
= ( ? [X: product_prod_v_v] :
( ( A1 = X )
& ( A22 = X ) )
| ? [X: product_prod_v_v,Y: product_prod_v_v,Z3: product_prod_v_v] :
( ( A1 = X )
& ( A22 = Z3 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Y )
& ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_427_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
= ( ? [X: v] :
( ( A1 = X )
& ( A22 = X ) )
| ? [X: v,Y: v,Z3: v] :
( ( A1 = X )
& ( A22 = Z3 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
& ( member_v @ Z3 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_428_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X3 @ X3 ) ) ).
% graph.re_refl
thf(fact_429_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X3: product_prod_v_v,Y4: product_prod_v_v,Z2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X3 @ Y4 )
=> ( ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y4 ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X3 @ Z2 ) ) ) ) ).
% graph.re_succ
thf(fact_430_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X3 @ Y4 )
=> ( ( member_v @ Z2 @ ( Successors @ Y4 ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X3 @ Z2 ) ) ) ) ).
% graph.re_succ
thf(fact_431_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_432_graph_Opre__dfss__explored__pre__dfss,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ Successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X: v] : ( if_set_v @ ( X = V ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X ) )
@ E ) ) ) ) ) ) ).
% graph.pre_dfss_explored_pre_dfss
thf(fact_433_graph_Osub__env__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ) ).
% graph.sub_env_trans
thf(fact_434_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_435_Collect__conv__if,axiom,
! [P: v > $o,A: v] :
( ( ( P @ A )
=> ( ( collect_v
@ ^ [X: v] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_v @ A @ bot_bot_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_v
@ ^ [X: v] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_v ) ) ) ).
% Collect_conv_if
thf(fact_436_Collect__conv__if,axiom,
! [P: product_prod_v_v > $o,A: product_prod_v_v] :
( ( ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] :
( ( X = A )
& ( P @ X ) ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
& ( ~ ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] :
( ( X = A )
& ( P @ X ) ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% Collect_conv_if
thf(fact_437_Collect__conv__if,axiom,
! [P: set_v > $o,A: set_v] :
( ( ( P @ A )
=> ( ( collect_set_v
@ ^ [X: set_v] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_v
@ ^ [X: set_v] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_set_v ) ) ) ).
% Collect_conv_if
thf(fact_438_Collect__conv__if2,axiom,
! [P: v > $o,A: v] :
( ( ( P @ A )
=> ( ( collect_v
@ ^ [X: v] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_v @ A @ bot_bot_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_v
@ ^ [X: v] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_v ) ) ) ).
% Collect_conv_if2
thf(fact_439_Collect__conv__if2,axiom,
! [P: product_prod_v_v > $o,A: product_prod_v_v] :
( ( ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] :
( ( A = X )
& ( P @ X ) ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
& ( ~ ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] :
( ( A = X )
& ( P @ X ) ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% Collect_conv_if2
thf(fact_440_Collect__conv__if2,axiom,
! [P: set_v > $o,A: set_v] :
( ( ( P @ A )
=> ( ( collect_set_v
@ ^ [X: set_v] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_v
@ ^ [X: set_v] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_set_v ) ) ) ).
% Collect_conv_if2
thf(fact_441_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_442_graph_Opre__dfss__unite__pre__dfss,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ 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 ) )
=> ( sCC_Bl3607325323686918683t_unit @ Successors @ V
@ ( sCC_Bl2958793191457503513t_unit
@ ^ [Uu: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v] : ( if_set4279007504652509325od_v_v @ ( X = V ) @ ( sup_su414716646722978715od_v_v @ ( sCC_Bl3878977043676959280t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) @ V ) @ ( insert1338601472111419319od_v_v @ W @ bot_bo723834152578015283od_v_v ) ) @ ( sCC_Bl3878977043676959280t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) @ X ) )
@ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ).
% graph.pre_dfss_unite_pre_dfss
thf(fact_443_graph_Opre__dfss__unite__pre__dfss,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,V: set_v,E: sCC_Bl337355980704484737t_unit,W: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7011918439528173327t_unit @ Successors @ V @ E )
=> ( ( member_set_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_set_v @ W @ ( sCC_Bl1947314421851746340t_unit @ E @ V ) )
=> ( ( member_set_v @ W @ ( sCC_Bl2616308362330240149t_unit @ E ) )
=> ( ~ ( member_set_v @ W @ ( sCC_Bl8056216847179350138t_unit @ E ) )
=> ( sCC_Bl7011918439528173327t_unit @ Successors @ V
@ ( sCC_Bl8982985682160534541t_unit
@ ^ [Uu: set_v > set_set_v,X: set_v] : ( if_set_set_v @ ( X = V ) @ ( sup_sup_set_set_v @ ( sCC_Bl1947314421851746340t_unit @ ( sCC_Bl2303001396873455969_set_v @ V @ W @ E ) @ V ) @ ( insert_set_v @ W @ bot_bot_set_set_v ) ) @ ( sCC_Bl1947314421851746340t_unit @ ( sCC_Bl2303001396873455969_set_v @ V @ W @ E ) @ X ) )
@ ( sCC_Bl2303001396873455969_set_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ).
% graph.pre_dfss_unite_pre_dfss
thf(fact_444_graph_Opre__dfss__unite__pre__dfss,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ 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 ) )
=> ( sCC_Bl1748261141445803503t_unit @ Successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X: v] : ( if_set_v @ ( X = V ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ V ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ X ) )
@ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ).
% graph.pre_dfss_unite_pre_dfss
thf(fact_445_graph_Opre__dfss__post__dfs__pre__dfss,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ( sCC_Bl5509015415614557509t_unit @ Successors @ W @ E @ ( sCC_Bl6251863795711659461od_v_v @ Successors @ W @ E ) )
=> ( sCC_Bl3607325323686918683t_unit @ Successors @ V
@ ( sCC_Bl2958793191457503513t_unit
@ ^ [Uu: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v] : ( if_set4279007504652509325od_v_v @ ( X = V ) @ ( sup_su414716646722978715od_v_v @ ( sCC_Bl3878977043676959280t_unit @ ( sCC_Bl6251863795711659461od_v_v @ Successors @ W @ E ) @ V ) @ ( insert1338601472111419319od_v_v @ W @ bot_bo723834152578015283od_v_v ) ) @ ( sCC_Bl3878977043676959280t_unit @ ( sCC_Bl6251863795711659461od_v_v @ Successors @ W @ E ) @ X ) )
@ ( sCC_Bl6251863795711659461od_v_v @ Successors @ W @ E ) ) ) ) ) ) ) ) ).
% graph.pre_dfss_post_dfs_pre_dfss
thf(fact_446_graph_Opre__dfss__post__dfs__pre__dfss,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,V: set_v,E: sCC_Bl337355980704484737t_unit,W: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7011918439528173327t_unit @ Successors @ V @ E )
=> ( ( member_set_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_set_v @ W @ ( sCC_Bl2616308362330240149t_unit @ E ) )
=> ( ( sCC_Bl6319204736534553937t_unit @ Successors @ W @ E @ ( sCC_Bl3939512277832569809_set_v @ Successors @ W @ E ) )
=> ( sCC_Bl7011918439528173327t_unit @ Successors @ V
@ ( sCC_Bl8982985682160534541t_unit
@ ^ [Uu: set_v > set_set_v,X: set_v] : ( if_set_set_v @ ( X = V ) @ ( sup_sup_set_set_v @ ( sCC_Bl1947314421851746340t_unit @ ( sCC_Bl3939512277832569809_set_v @ Successors @ W @ E ) @ V ) @ ( insert_set_v @ W @ bot_bot_set_set_v ) ) @ ( sCC_Bl1947314421851746340t_unit @ ( sCC_Bl3939512277832569809_set_v @ Successors @ W @ E ) @ X ) )
@ ( sCC_Bl3939512277832569809_set_v @ Successors @ W @ E ) ) ) ) ) ) ) ) ).
% graph.pre_dfss_post_dfs_pre_dfss
thf(fact_447_graph_Opre__dfss__post__dfs__pre__dfss,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ W @ E @ ( sCC_Bloemen_dfs_v @ Successors @ W @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ Successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X: v] : ( if_set_v @ ( X = V ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ Successors @ W @ E ) @ V ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ Successors @ W @ E ) @ X ) )
@ ( sCC_Bloemen_dfs_v @ Successors @ W @ E ) ) ) ) ) ) ) ) ).
% graph.pre_dfss_post_dfs_pre_dfss
thf(fact_448_Diff__insert__absorb,axiom,
! [X3: product_prod_v_v,A2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X3 @ A2 ) @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_449_Diff__insert__absorb,axiom,
! [X3: set_v,A2: set_set_v] :
( ~ ( member_set_v @ X3 @ A2 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X3 @ A2 ) @ ( insert_set_v @ X3 @ bot_bot_set_set_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_450_Diff__insert__absorb,axiom,
! [X3: v,A2: set_v] :
( ~ ( member_v @ X3 @ A2 )
=> ( ( minus_minus_set_v @ ( insert_v @ X3 @ A2 ) @ ( insert_v @ X3 @ bot_bot_set_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_451_Diff__insert2,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_452_Diff__insert2,axiom,
! [A2: set_set_v,A: set_v,B: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_453_Diff__insert2,axiom,
! [A2: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v @ A @ bot_bot_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_454_insert__Diff,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_455_insert__Diff,axiom,
! [A: set_v,A2: set_set_v] :
( ( member_set_v @ A @ A2 )
=> ( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_456_insert__Diff,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( insert_v @ A @ ( minus_minus_set_v @ A2 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_457_Diff__insert,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_458_Diff__insert,axiom,
! [A2: set_set_v,A: set_v,B: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B ) @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ).
% Diff_insert
thf(fact_459_Diff__insert,axiom,
! [A2: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ B ) @ ( insert_v @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_460_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_461_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X3 @ Y4 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y4 ) ) ) ).
% graph.re_reachable
thf(fact_462_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y4 )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X3 @ Y4 ) ) ) ).
% graph.reachable_re
thf(fact_463_unfold__congs_I5_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V2: v > set_v,F: ( v > set_v ) > v > set_v,F2: ( v > set_v ) > v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R2 )
= V2 )
=> ( ! [V3: v > set_v] :
( ( V3 = V2 )
=> ( ( F @ V3 )
= ( F2 @ V3 ) ) )
=> ( ( sCC_Bl48393358579903213t_unit @ F @ R )
= ( sCC_Bl48393358579903213t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(5)
thf(fact_464_fold__congs_I5_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V2: v > set_v,F: ( v > set_v ) > v > set_v,F2: ( v > set_v ) > v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R2 )
= V2 )
=> ( ! [V3: v > set_v] :
( ( V2 = V3 )
=> ( ( F @ V3 )
= ( F2 @ V3 ) ) )
=> ( ( sCC_Bl48393358579903213t_unit @ F @ R )
= ( sCC_Bl48393358579903213t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(5)
thf(fact_465_graph_Opre__dfss__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ W @ E ) ) ) ) ) ).
% graph.pre_dfss_pre_dfs
thf(fact_466_verit__sko__forall__indirect2,axiom,
! [X3: v,P: v > $o,P5: v > $o] :
( ( X3
= ( fChoice_v
@ ^ [X: v] :
~ ( P @ X ) ) )
=> ( ! [X2: v] :
( ( P @ X2 )
= ( P5 @ X2 ) )
=> ( ( ! [X5: v] : ( P5 @ X5 ) )
= ( P @ X3 ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_467_verit__sko__forall__indirect,axiom,
! [X3: v,P: v > $o] :
( ( X3
= ( fChoice_v
@ ^ [X: v] :
~ ( P @ X ) ) )
=> ( ( ! [X5: v] : ( P @ X5 ) )
= ( P @ X3 ) ) ) ).
% verit_sko_forall_indirect
thf(fact_468_verit__sko__ex__indirect2,axiom,
! [X3: v,P: v > $o,P5: v > $o] :
( ( X3
= ( fChoice_v @ P ) )
=> ( ! [X2: v] :
( ( P @ X2 )
= ( P5 @ X2 ) )
=> ( ( ? [X5: v] : ( P5 @ X5 ) )
= ( P @ X3 ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_469_verit__sko__ex__indirect,axiom,
! [X3: v,P: v > $o] :
( ( X3
= ( fChoice_v @ P ) )
=> ( ( ? [X5: v] : ( P @ X5 ) )
= ( P @ X3 ) ) ) ).
% verit_sko_ex_indirect
thf(fact_470_verit__sko__forall_H_H,axiom,
! [B: v,A2: v,P: v > $o] :
( ( B = A2 )
=> ( ( ( fChoice_v @ P )
= A2 )
= ( ( fChoice_v @ P )
= B ) ) ) ).
% verit_sko_forall''
thf(fact_471_verit__sko__forall_H,axiom,
! [P: v > $o,A2: $o] :
( ( ( P
@ ( fChoice_v
@ ^ [X: v] :
~ ( P @ X ) ) )
= A2 )
=> ( ( ! [X5: v] : ( P @ X5 ) )
= A2 ) ) ).
% verit_sko_forall'
thf(fact_472_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 )
& ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ V ) ) ) ) ).
% pre_dfs_def
thf(fact_473_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_474_unite__subscc,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ 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 ) )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ).
% unite_subscc
thf(fact_475_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_476_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 )
& ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ V ) ) ) ) ) ).
% graph.pre_dfs_def
thf(fact_477_dfs__S__tl__stack_I1_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ).
% dfs_S_tl_stack(1)
thf(fact_478_scc__partition,axiom,
! [S: set_v,S3: set_v,X3: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( member_v @ X3 @ ( inf_inf_set_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ).
% scc_partition
thf(fact_479_the__elem__eq,axiom,
! [X3: v] :
( ( the_elem_v @ ( insert_v @ X3 @ bot_bot_set_v ) )
= X3 ) ).
% the_elem_eq
thf(fact_480_the__elem__eq,axiom,
! [X3: product_prod_v_v] :
( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
= X3 ) ).
% the_elem_eq
thf(fact_481_the__elem__eq,axiom,
! [X3: set_v] :
( ( the_elem_set_v @ ( insert_set_v @ X3 @ bot_bot_set_set_v ) )
= X3 ) ).
% the_elem_eq
thf(fact_482_ra__mono,axiom,
! [X3: v,Y4: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y4 @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y4 @ E5 ) ) ) ).
% ra_mono
thf(fact_483_bot__empty__eq,axiom,
( bot_bot_v_o
= ( ^ [X: v] : ( member_v @ X @ bot_bot_set_v ) ) ) ).
% bot_empty_eq
thf(fact_484_bot__empty__eq,axiom,
( bot_bo8461541820394803818_v_v_o
= ( ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ).
% bot_empty_eq
thf(fact_485_bot__empty__eq,axiom,
( bot_bot_set_v_o
= ( ^ [X: set_v] : ( member_set_v @ X @ bot_bot_set_set_v ) ) ) ).
% bot_empty_eq
thf(fact_486_order__refl,axiom,
! [X3: set_v] : ( ord_less_eq_set_v @ X3 @ X3 ) ).
% order_refl
thf(fact_487_order__refl,axiom,
! [X3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X3 @ X3 ) ).
% order_refl
thf(fact_488_dual__order_Orefl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% dual_order.refl
thf(fact_489_dual__order_Orefl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% dual_order.refl
thf(fact_490_subsetI,axiom,
! [A2: set_v,B: set_v] :
( ! [X2: v] :
( ( member_v @ X2 @ A2 )
=> ( member_v @ X2 @ B ) )
=> ( ord_less_eq_set_v @ A2 @ B ) ) ).
% subsetI
thf(fact_491_subsetI,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
=> ( member7453568604450474000od_v_v @ X2 @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B ) ) ).
% subsetI
thf(fact_492_subset__antisym,axiom,
! [A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( ord_less_eq_set_v @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_493_subset__antisym,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_494_inf_Oidem,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% inf.idem
thf(fact_495_inf_Oidem,axiom,
! [A: product_unit] :
( ( inf_inf_Product_unit @ A @ A )
= A ) ).
% inf.idem
thf(fact_496_inf__idem,axiom,
! [X3: set_v] :
( ( inf_inf_set_v @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_497_inf__idem,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_498_inf_Oleft__idem,axiom,
! [A: set_v,B3: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B3 ) )
= ( inf_inf_set_v @ A @ B3 ) ) ).
% inf.left_idem
thf(fact_499_inf_Oleft__idem,axiom,
! [A: product_unit,B3: product_unit] :
( ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ A @ B3 ) )
= ( inf_inf_Product_unit @ A @ B3 ) ) ).
% inf.left_idem
thf(fact_500_inf__left__idem,axiom,
! [X3: set_v,Y4: set_v] :
( ( inf_inf_set_v @ X3 @ ( inf_inf_set_v @ X3 @ Y4 ) )
= ( inf_inf_set_v @ X3 @ Y4 ) ) ).
% inf_left_idem
thf(fact_501_inf__left__idem,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( inf_inf_Product_unit @ X3 @ Y4 ) )
= ( inf_inf_Product_unit @ X3 @ Y4 ) ) ).
% inf_left_idem
thf(fact_502_inf_Oright__idem,axiom,
! [A: set_v,B3: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B3 ) @ B3 )
= ( inf_inf_set_v @ A @ B3 ) ) ).
% inf.right_idem
thf(fact_503_inf_Oright__idem,axiom,
! [A: product_unit,B3: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A @ B3 ) @ B3 )
= ( inf_inf_Product_unit @ A @ B3 ) ) ).
% inf.right_idem
thf(fact_504_inf__right__idem,axiom,
! [X3: set_v,Y4: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X3 @ Y4 ) @ Y4 )
= ( inf_inf_set_v @ X3 @ Y4 ) ) ).
% inf_right_idem
thf(fact_505_inf__right__idem,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ Y4 )
= ( inf_inf_Product_unit @ X3 @ Y4 ) ) ).
% inf_right_idem
thf(fact_506_IntI,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_507_IntI,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ A2 )
=> ( ( member_v @ C @ B )
=> ( member_v @ C @ ( inf_inf_set_v @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_508_Int__iff,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A2 )
& ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_509_Int__iff,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B ) )
= ( ( member_v @ C @ A2 )
& ( member_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_510_sclosed,axiom,
! [X4: v] :
( ( member_v @ X4 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X4 ) @ vertices ) ) ).
% sclosed
thf(fact_511_le__inf__iff,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) )
= ( ( ord_le3221252021190050221t_unit @ X3 @ Y4 )
& ( ord_le3221252021190050221t_unit @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_512_le__inf__iff,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ X3 @ ( inf_inf_set_v @ Y4 @ Z2 ) )
= ( ( ord_less_eq_set_v @ X3 @ Y4 )
& ( ord_less_eq_set_v @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_513_le__inf__iff,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y4 @ Z2 ) )
= ( ( ord_le7336532860387713383od_v_v @ X3 @ Y4 )
& ( ord_le7336532860387713383od_v_v @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_514_inf_Obounded__iff,axiom,
! [A: product_unit,B3: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B3 @ C ) )
= ( ( ord_le3221252021190050221t_unit @ A @ B3 )
& ( ord_le3221252021190050221t_unit @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_515_inf_Obounded__iff,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B3 @ C ) )
= ( ( ord_less_eq_set_v @ A @ B3 )
& ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_516_inf_Obounded__iff,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B3 @ C ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ B3 )
& ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_517_le__sup__iff,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X3 @ Y4 ) @ Z2 )
= ( ( ord_le5216385588623774835_set_v @ X3 @ Z2 )
& ( ord_le5216385588623774835_set_v @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_518_le__sup__iff,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ Z2 )
= ( ( ord_le3221252021190050221t_unit @ X3 @ Z2 )
& ( ord_le3221252021190050221t_unit @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_519_le__sup__iff,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X3 @ Y4 ) @ Z2 )
= ( ( ord_less_eq_set_v @ X3 @ Z2 )
& ( ord_less_eq_set_v @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_520_le__sup__iff,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) @ Z2 )
= ( ( ord_le7336532860387713383od_v_v @ X3 @ Z2 )
& ( ord_le7336532860387713383od_v_v @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_521_sup_Obounded__iff,axiom,
! [B3: set_set_v,C: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B3 @ C ) @ A )
= ( ( ord_le5216385588623774835_set_v @ B3 @ A )
& ( ord_le5216385588623774835_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_522_sup_Obounded__iff,axiom,
! [B3: product_unit,C: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B3 @ C ) @ A )
= ( ( ord_le3221252021190050221t_unit @ B3 @ A )
& ( ord_le3221252021190050221t_unit @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_523_sup_Obounded__iff,axiom,
! [B3: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B3 @ C ) @ A )
= ( ( ord_less_eq_set_v @ B3 @ A )
& ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_524_sup_Obounded__iff,axiom,
! [B3: 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 @ B3 @ C ) @ A )
= ( ( ord_le7336532860387713383od_v_v @ B3 @ A )
& ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_525_empty__subsetI,axiom,
! [A2: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A2 ) ).
% empty_subsetI
thf(fact_526_empty__subsetI,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A2 ) ).
% empty_subsetI
thf(fact_527_empty__subsetI,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A2 ) ).
% empty_subsetI
thf(fact_528_subset__empty,axiom,
! [A2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ bot_bot_set_set_v )
= ( A2 = bot_bot_set_set_v ) ) ).
% subset_empty
thf(fact_529_subset__empty,axiom,
! [A2: set_v] :
( ( ord_less_eq_set_v @ A2 @ bot_bot_set_v )
= ( A2 = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_530_subset__empty,axiom,
! [A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_531_inf__bot__right,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ X3 @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% inf_bot_right
thf(fact_532_inf__bot__right,axiom,
! [X3: set_v] :
( ( inf_inf_set_v @ X3 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% inf_bot_right
thf(fact_533_inf__bot__right,axiom,
! [X3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_right
thf(fact_534_inf__bot__right,axiom,
! [X3: set_set_v] :
( ( inf_inf_set_set_v @ X3 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% inf_bot_right
thf(fact_535_inf__bot__left,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X3 )
= bot_bot_Product_unit ) ).
% inf_bot_left
thf(fact_536_inf__bot__left,axiom,
! [X3: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X3 )
= bot_bot_set_v ) ).
% inf_bot_left
thf(fact_537_inf__bot__left,axiom,
! [X3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X3 )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_left
thf(fact_538_inf__bot__left,axiom,
! [X3: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X3 )
= bot_bot_set_set_v ) ).
% inf_bot_left
thf(fact_539_insert__subset,axiom,
! [X3: set_v,A2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( insert_set_v @ X3 @ A2 ) @ B )
= ( ( member_set_v @ X3 @ B )
& ( ord_le5216385588623774835_set_v @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_540_insert__subset,axiom,
! [X3: v,A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ ( insert_v @ X3 @ A2 ) @ B )
= ( ( member_v @ X3 @ B )
& ( ord_less_eq_set_v @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_541_insert__subset,axiom,
! [X3: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X3 @ A2 ) @ B )
= ( ( member7453568604450474000od_v_v @ X3 @ B )
& ( ord_le7336532860387713383od_v_v @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_542_inf__sup__absorb,axiom,
! [X3: set_v,Y4: set_v] :
( ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ X3 @ Y4 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_543_inf__sup__absorb,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_544_inf__sup__absorb,axiom,
! [X3: set_set_v,Y4: set_set_v] :
( ( inf_inf_set_set_v @ X3 @ ( sup_sup_set_set_v @ X3 @ Y4 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_545_inf__sup__absorb,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( sup_sup_Product_unit @ X3 @ Y4 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_546_sup__inf__absorb,axiom,
! [X3: set_v,Y4: set_v] :
( ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ X3 @ Y4 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_547_sup__inf__absorb,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ X3 @ Y4 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_548_sup__inf__absorb,axiom,
! [X3: set_set_v,Y4: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ ( inf_inf_set_set_v @ X3 @ Y4 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_549_sup__inf__absorb,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( inf_inf_Product_unit @ X3 @ Y4 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_550_Int__subset__iff,axiom,
! [C2: set_v,A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A2 @ B ) )
= ( ( ord_less_eq_set_v @ C2 @ A2 )
& ( ord_less_eq_set_v @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_551_Int__subset__iff,axiom,
! [C2: set_Product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) )
= ( ( ord_le7336532860387713383od_v_v @ C2 @ A2 )
& ( ord_le7336532860387713383od_v_v @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_552_Un__subset__iff,axiom,
! [A2: set_set_v,B: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ C2 )
= ( ( ord_le5216385588623774835_set_v @ A2 @ C2 )
& ( ord_le5216385588623774835_set_v @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_553_Un__subset__iff,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B ) @ C2 )
= ( ( ord_less_eq_set_v @ A2 @ C2 )
& ( ord_less_eq_set_v @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_554_Un__subset__iff,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ C2 )
= ( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
& ( ord_le7336532860387713383od_v_v @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_555_Int__insert__left__if0,axiom,
! [A: set_v,C2: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C2 )
= ( inf_inf_set_set_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_556_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_557_Int__insert__left__if0,axiom,
! [A: v,C2: set_v,B: set_v] :
( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C2 )
= ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_558_Int__insert__left__if1,axiom,
! [A: set_v,C2: set_set_v,B: set_set_v] :
( ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C2 )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_559_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_560_Int__insert__left__if1,axiom,
! [A: v,C2: set_v,B: set_v] :
( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C2 )
= ( insert_v @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_561_insert__inter__insert,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_562_insert__inter__insert,axiom,
! [A: set_v,A2: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A2 ) @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_563_insert__inter__insert,axiom,
! [A: v,A2: set_v,B: set_v] :
( ( inf_inf_set_v @ ( insert_v @ A @ A2 ) @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_564_Int__insert__right__if0,axiom,
! [A: set_v,A2: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ A @ A2 )
=> ( ( inf_inf_set_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( inf_inf_set_set_v @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_565_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_566_Int__insert__right__if0,axiom,
! [A: v,A2: set_v,B: set_v] :
( ~ ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( inf_inf_set_v @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_567_Int__insert__right__if1,axiom,
! [A: set_v,A2: set_set_v,B: set_set_v] :
( ( member_set_v @ A @ A2 )
=> ( ( inf_inf_set_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_568_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_569_Int__insert__right__if1,axiom,
! [A: v,A2: set_v,B: set_v] :
( ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_570_Int__Un__eq_I4_J,axiom,
! [T: set_v,S: set_v] :
( ( sup_sup_set_v @ T @ ( inf_inf_set_v @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_571_Int__Un__eq_I4_J,axiom,
! [T: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T @ ( inf_in6271465464967711157od_v_v @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_572_Int__Un__eq_I4_J,axiom,
! [T: set_set_v,S: set_set_v] :
( ( sup_sup_set_set_v @ T @ ( inf_inf_set_set_v @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_573_Int__Un__eq_I3_J,axiom,
! [S: set_v,T: set_v] :
( ( sup_sup_set_v @ S @ ( inf_inf_set_v @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_574_Int__Un__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S @ ( inf_in6271465464967711157od_v_v @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_575_Int__Un__eq_I3_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( sup_sup_set_set_v @ S @ ( inf_inf_set_set_v @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_576_Int__Un__eq_I2_J,axiom,
! [S: set_v,T: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_577_Int__Un__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_578_Int__Un__eq_I2_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_579_Int__Un__eq_I1_J,axiom,
! [S: set_v,T: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_580_Int__Un__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_581_Int__Un__eq_I1_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_582_Un__Int__eq_I4_J,axiom,
! [T: set_v,S: set_v] :
( ( inf_inf_set_v @ T @ ( sup_sup_set_v @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_583_Un__Int__eq_I4_J,axiom,
! [T: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T @ ( sup_su414716646722978715od_v_v @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_584_Un__Int__eq_I4_J,axiom,
! [T: set_set_v,S: set_set_v] :
( ( inf_inf_set_set_v @ T @ ( sup_sup_set_set_v @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_585_Un__Int__eq_I3_J,axiom,
! [S: set_v,T: set_v] :
( ( inf_inf_set_v @ S @ ( sup_sup_set_v @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_586_Un__Int__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S @ ( sup_su414716646722978715od_v_v @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_587_Un__Int__eq_I3_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( inf_inf_set_set_v @ S @ ( sup_sup_set_set_v @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_588_Un__Int__eq_I2_J,axiom,
! [S: set_v,T: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_589_Un__Int__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_590_Un__Int__eq_I2_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_591_Un__Int__eq_I1_J,axiom,
! [S: set_v,T: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_592_Un__Int__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_593_Un__Int__eq_I1_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_594_dfs__S__hd__stack_I1_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ).
% dfs_S_hd_stack(1)
thf(fact_595_dfs__S__hd__stack_I2_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( 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 @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ).
% dfs_S_hd_stack(2)
thf(fact_596_dfs__S__tl__stack_I2_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) ) ).
% dfs_S_tl_stack(2)
thf(fact_597_singleton__insert__inj__eq_H,axiom,
! [A: set_v,A2: set_set_v,B3: set_v] :
( ( ( insert_set_v @ A @ A2 )
= ( insert_set_v @ B3 @ bot_bot_set_set_v ) )
= ( ( A = B3 )
& ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ B3 @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_598_singleton__insert__inj__eq_H,axiom,
! [A: v,A2: set_v,B3: v] :
( ( ( insert_v @ A @ A2 )
= ( insert_v @ B3 @ bot_bot_set_v ) )
= ( ( A = B3 )
& ( ord_less_eq_set_v @ A2 @ ( insert_v @ B3 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_599_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B3: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A2 )
= ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B3 )
& ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_600_singleton__insert__inj__eq,axiom,
! [B3: set_v,A: set_v,A2: set_set_v] :
( ( ( insert_set_v @ B3 @ bot_bot_set_set_v )
= ( insert_set_v @ A @ A2 ) )
= ( ( A = B3 )
& ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ B3 @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_601_singleton__insert__inj__eq,axiom,
! [B3: v,A: v,A2: set_v] :
( ( ( insert_v @ B3 @ bot_bot_set_v )
= ( insert_v @ A @ A2 ) )
= ( ( A = B3 )
& ( ord_less_eq_set_v @ A2 @ ( insert_v @ B3 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_602_singleton__insert__inj__eq,axiom,
! [B3: product_prod_v_v,A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A2 ) )
= ( ( A = B3 )
& ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_603_insert__disjoint_I1_J,axiom,
! [A: v,A2: set_v,B: set_v] :
( ( ( inf_inf_set_v @ ( insert_v @ A @ A2 ) @ B )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B )
& ( ( inf_inf_set_v @ A2 @ B )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_604_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ B )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_605_insert__disjoint_I1_J,axiom,
! [A: set_v,A2: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A2 ) @ B )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B )
& ( ( inf_inf_set_set_v @ A2 @ B )
= bot_bot_set_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_606_insert__disjoint_I2_J,axiom,
! [A: v,A2: set_v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v @ A @ A2 ) @ B ) )
= ( ~ ( member_v @ A @ B )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_607_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ B ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_608_insert__disjoint_I2_J,axiom,
! [A: set_v,A2: set_set_v,B: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ ( insert_set_v @ A @ A2 ) @ B ) )
= ( ~ ( member_set_v @ A @ B )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_609_disjoint__insert_I1_J,axiom,
! [B: set_v,A: v,A2: set_v] :
( ( ( inf_inf_set_v @ B @ ( insert_v @ A @ A2 ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B )
& ( ( inf_inf_set_v @ B @ A2 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_610_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ A2 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ B @ A2 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_611_disjoint__insert_I1_J,axiom,
! [B: set_set_v,A: set_v,A2: set_set_v] :
( ( ( inf_inf_set_set_v @ B @ ( insert_set_v @ A @ A2 ) )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B )
& ( ( inf_inf_set_set_v @ B @ A2 )
= bot_bot_set_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_612_disjoint__insert_I2_J,axiom,
! [A2: set_v,B3: v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ ( insert_v @ B3 @ B ) ) )
= ( ~ ( member_v @ B3 @ A2 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_613_disjoint__insert_I2_J,axiom,
! [A2: set_Product_prod_v_v,B3: product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B3 @ B ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B3 @ A2 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_614_disjoint__insert_I2_J,axiom,
! [A2: set_set_v,B3: set_v,B: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A2 @ ( insert_set_v @ B3 @ B ) ) )
= ( ~ ( member_set_v @ B3 @ A2 )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_615_Diff__eq__empty__iff,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ A2 @ B )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_616_Diff__eq__empty__iff,axiom,
! [A2: set_v,B: set_v] :
( ( ( minus_minus_set_v @ A2 @ B )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_617_Diff__eq__empty__iff,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_618_Diff__disjoint,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B @ A2 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_619_Diff__disjoint,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B @ A2 ) )
= bot_bot_set_set_v ) ).
% Diff_disjoint
thf(fact_620_Diff__disjoint,axiom,
! [A2: set_v,B: set_v] :
( ( inf_inf_set_v @ A2 @ ( minus_minus_set_v @ B @ A2 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_621_verit__comp__simplify1_I2_J,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_622_verit__comp__simplify1_I2_J,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_623_Un__Int__assoc__eq,axiom,
! [A2: set_set_v,B: set_set_v,C2: set_set_v] :
( ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_v @ A2 @ ( sup_sup_set_set_v @ B @ C2 ) ) )
= ( ord_le5216385588623774835_set_v @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_624_Un__Int__assoc__eq,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B ) @ C2 )
= ( inf_inf_set_v @ A2 @ ( sup_sup_set_v @ B @ C2 ) ) )
= ( ord_less_eq_set_v @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_625_Un__Int__assoc__eq,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) )
= ( ord_le7336532860387713383od_v_v @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_626_distrib__sup__le,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X3 @ ( inf_inf_set_set_v @ Y4 @ Z2 ) ) @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X3 @ Y4 ) @ ( sup_sup_set_set_v @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_627_distrib__sup__le,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) ) @ ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_628_distrib__sup__le,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ Y4 @ Z2 ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X3 @ Y4 ) @ ( sup_sup_set_v @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_629_distrib__sup__le,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y4 @ Z2 ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) @ ( sup_su414716646722978715od_v_v @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_630_distrib__inf__le,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X3 @ Y4 ) @ ( inf_inf_set_set_v @ X3 @ Z2 ) ) @ ( inf_inf_set_set_v @ X3 @ ( sup_sup_set_set_v @ Y4 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_631_distrib__inf__le,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) @ ( inf_inf_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_632_distrib__inf__le,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X3 @ Y4 ) @ ( inf_inf_set_v @ X3 @ Z2 ) ) @ ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ Y4 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_633_distrib__inf__le,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y4 ) @ ( inf_in6271465464967711157od_v_v @ X3 @ Z2 ) ) @ ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y4 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_634_inf__sup__aci_I4_J,axiom,
! [X3: set_v,Y4: set_v] :
( ( inf_inf_set_v @ X3 @ ( inf_inf_set_v @ X3 @ Y4 ) )
= ( inf_inf_set_v @ X3 @ Y4 ) ) ).
% inf_sup_aci(4)
thf(fact_635_inf__sup__aci_I4_J,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( inf_inf_Product_unit @ X3 @ Y4 ) )
= ( inf_inf_Product_unit @ X3 @ Y4 ) ) ).
% inf_sup_aci(4)
thf(fact_636_inf__sup__aci_I3_J,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( inf_inf_set_v @ X3 @ ( inf_inf_set_v @ Y4 @ Z2 ) )
= ( inf_inf_set_v @ Y4 @ ( inf_inf_set_v @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_637_inf__sup__aci_I3_J,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) )
= ( inf_inf_Product_unit @ Y4 @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_638_inf__sup__aci_I2_J,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X3 @ Y4 ) @ Z2 )
= ( inf_inf_set_v @ X3 @ ( inf_inf_set_v @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_639_inf__sup__aci_I2_J,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ Z2 )
= ( inf_inf_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_640_inf__sup__aci_I1_J,axiom,
( inf_inf_set_v
= ( ^ [X: set_v,Y: set_v] : ( inf_inf_set_v @ Y @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_641_inf__sup__aci_I1_J,axiom,
( inf_inf_Product_unit
= ( ^ [X: product_unit,Y: product_unit] : ( inf_inf_Product_unit @ Y @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_642_inf__sup__ord_I2_J,axiom,
! [X3: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_643_inf__sup__ord_I2_J,axiom,
! [X3: set_v,Y4: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X3 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_644_inf__sup__ord_I2_J,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_645_inf__sup__ord_I1_J,axiom,
! [X3: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_646_inf__sup__ord_I1_J,axiom,
! [X3: set_v,Y4: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X3 @ Y4 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_647_inf__sup__ord_I1_J,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y4 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_648_inf__le1,axiom,
! [X3: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ X3 ) ).
% inf_le1
thf(fact_649_inf__le1,axiom,
! [X3: set_v,Y4: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X3 @ Y4 ) @ X3 ) ).
% inf_le1
thf(fact_650_inf__le1,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y4 ) @ X3 ) ).
% inf_le1
thf(fact_651_inf__le2,axiom,
! [X3: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_652_inf__le2,axiom,
! [X3: set_v,Y4: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X3 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_653_inf__le2,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_654_le__infE,axiom,
! [X3: product_unit,A: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ ( inf_inf_Product_unit @ A @ B3 ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ X3 @ A )
=> ~ ( ord_le3221252021190050221t_unit @ X3 @ B3 ) ) ) ).
% le_infE
thf(fact_655_le__infE,axiom,
! [X3: set_v,A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ X3 @ ( inf_inf_set_v @ A @ B3 ) )
=> ~ ( ( ord_less_eq_set_v @ X3 @ A )
=> ~ ( ord_less_eq_set_v @ X3 @ B3 ) ) ) ).
% le_infE
thf(fact_656_le__infE,axiom,
! [X3: set_Product_prod_v_v,A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X3 @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ X3 @ B3 ) ) ) ).
% le_infE
thf(fact_657_le__infI,axiom,
! [X3: product_unit,A: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ A )
=> ( ( ord_le3221252021190050221t_unit @ X3 @ B3 )
=> ( ord_le3221252021190050221t_unit @ X3 @ ( inf_inf_Product_unit @ A @ B3 ) ) ) ) ).
% le_infI
thf(fact_658_le__infI,axiom,
! [X3: set_v,A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ X3 @ A )
=> ( ( ord_less_eq_set_v @ X3 @ B3 )
=> ( ord_less_eq_set_v @ X3 @ ( inf_inf_set_v @ A @ B3 ) ) ) ) ).
% le_infI
thf(fact_659_le__infI,axiom,
! [X3: set_Product_prod_v_v,A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ B3 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) ) ) ) ).
% le_infI
thf(fact_660_inf__mono,axiom,
! [A: product_unit,C: product_unit,B3: product_unit,D: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ( ord_le3221252021190050221t_unit @ B3 @ D )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B3 ) @ ( inf_inf_Product_unit @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_661_inf__mono,axiom,
! [A: set_v,C: set_v,B3: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B3 @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ ( inf_inf_set_v @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_662_inf__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B3: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ ( inf_in6271465464967711157od_v_v @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_663_le__infI1,axiom,
! [A: product_unit,X3: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ X3 )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B3 ) @ X3 ) ) ).
% le_infI1
thf(fact_664_le__infI1,axiom,
! [A: set_v,X3: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ X3 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ X3 ) ) ).
% le_infI1
thf(fact_665_le__infI1,axiom,
! [A: set_Product_prod_v_v,X3: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X3 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ X3 ) ) ).
% le_infI1
thf(fact_666_le__infI2,axiom,
! [B3: product_unit,X3: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B3 @ X3 )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B3 ) @ X3 ) ) ).
% le_infI2
thf(fact_667_le__infI2,axiom,
! [B3: set_v,X3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B3 @ X3 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ X3 ) ) ).
% le_infI2
thf(fact_668_le__infI2,axiom,
! [B3: set_Product_prod_v_v,X3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ X3 ) ) ).
% le_infI2
thf(fact_669_inf_Oassoc,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B3 ) @ C )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_670_inf_Oassoc,axiom,
! [A: product_unit,B3: product_unit,C: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A @ B3 ) @ C )
= ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_671_inf__assoc,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X3 @ Y4 ) @ Z2 )
= ( inf_inf_set_v @ X3 @ ( inf_inf_set_v @ Y4 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_672_inf__assoc,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ Z2 )
= ( inf_inf_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_673_inf_OorderE,axiom,
! [A: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B3 )
=> ( A
= ( inf_inf_Product_unit @ A @ B3 ) ) ) ).
% inf.orderE
thf(fact_674_inf_OorderE,axiom,
! [A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( A
= ( inf_inf_set_v @ A @ B3 ) ) ) ).
% inf.orderE
thf(fact_675_inf_OorderE,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( A
= ( inf_in6271465464967711157od_v_v @ A @ B3 ) ) ) ).
% inf.orderE
thf(fact_676_inf_OorderI,axiom,
! [A: product_unit,B3: product_unit] :
( ( A
= ( inf_inf_Product_unit @ A @ B3 ) )
=> ( ord_le3221252021190050221t_unit @ A @ B3 ) ) ).
% inf.orderI
thf(fact_677_inf_OorderI,axiom,
! [A: set_v,B3: set_v] :
( ( A
= ( inf_inf_set_v @ A @ B3 ) )
=> ( ord_less_eq_set_v @ A @ B3 ) ) ).
% inf.orderI
thf(fact_678_inf_OorderI,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( A
= ( inf_in6271465464967711157od_v_v @ A @ B3 ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B3 ) ) ).
% inf.orderI
thf(fact_679_inf__unique,axiom,
! [F: product_unit > product_unit > product_unit,X3: product_unit,Y4: product_unit] :
( ! [X2: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: product_unit,Y3: product_unit,Z4: product_unit] :
( ( ord_le3221252021190050221t_unit @ X2 @ Y3 )
=> ( ( ord_le3221252021190050221t_unit @ X2 @ Z4 )
=> ( ord_le3221252021190050221t_unit @ X2 @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_inf_Product_unit @ X3 @ Y4 )
= ( F @ X3 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_680_inf__unique,axiom,
! [F: set_v > set_v > set_v,X3: set_v,Y4: set_v] :
( ! [X2: set_v,Y3: set_v] : ( ord_less_eq_set_v @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: set_v,Y3: set_v] : ( ord_less_eq_set_v @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: set_v,Y3: set_v,Z4: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
=> ( ( ord_less_eq_set_v @ X2 @ Z4 )
=> ( ord_less_eq_set_v @ X2 @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_inf_set_v @ X3 @ Y4 )
= ( F @ X3 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_681_inf__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v,Z4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
=> ( ( ord_le7336532860387713383od_v_v @ X2 @ Z4 )
=> ( ord_le7336532860387713383od_v_v @ X2 @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X3 @ Y4 )
= ( F @ X3 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_682_le__iff__inf,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_683_le__iff__inf,axiom,
( ord_less_eq_set_v
= ( ^ [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_684_le__iff__inf,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_685_inf_Oabsorb1,axiom,
! [A: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B3 )
=> ( ( inf_inf_Product_unit @ A @ B3 )
= A ) ) ).
% inf.absorb1
thf(fact_686_inf_Oabsorb1,axiom,
! [A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( inf_inf_set_v @ A @ B3 )
= A ) ) ).
% inf.absorb1
thf(fact_687_inf_Oabsorb1,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B3 )
= A ) ) ).
% inf.absorb1
thf(fact_688_inf_Oabsorb2,axiom,
! [B3: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B3 @ A )
=> ( ( inf_inf_Product_unit @ A @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_689_inf_Oabsorb2,axiom,
! [B3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B3 @ A )
=> ( ( inf_inf_set_v @ A @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_690_inf_Oabsorb2,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_691_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B5: set_v] : ( inf_inf_set_v @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_692_inf_Ocommute,axiom,
( inf_inf_Product_unit
= ( ^ [A4: product_unit,B5: product_unit] : ( inf_inf_Product_unit @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_693_inf__absorb1,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ Y4 )
=> ( ( inf_inf_Product_unit @ X3 @ Y4 )
= X3 ) ) ).
% inf_absorb1
thf(fact_694_inf__absorb1,axiom,
! [X3: set_v,Y4: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y4 )
=> ( ( inf_inf_set_v @ X3 @ Y4 )
= X3 ) ) ).
% inf_absorb1
thf(fact_695_inf__absorb1,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y4 )
=> ( ( inf_in6271465464967711157od_v_v @ X3 @ Y4 )
= X3 ) ) ).
% inf_absorb1
thf(fact_696_inf__absorb2,axiom,
! [Y4: product_unit,X3: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y4 @ X3 )
=> ( ( inf_inf_Product_unit @ X3 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_697_inf__absorb2,axiom,
! [Y4: set_v,X3: set_v] :
( ( ord_less_eq_set_v @ Y4 @ X3 )
=> ( ( inf_inf_set_v @ X3 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_698_inf__absorb2,axiom,
! [Y4: set_Product_prod_v_v,X3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y4 @ X3 )
=> ( ( inf_in6271465464967711157od_v_v @ X3 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_699_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X: set_v,Y: set_v] : ( inf_inf_set_v @ Y @ X ) ) ) ).
% inf_commute
thf(fact_700_inf__commute,axiom,
( inf_inf_Product_unit
= ( ^ [X: product_unit,Y: product_unit] : ( inf_inf_Product_unit @ Y @ X ) ) ) ).
% inf_commute
thf(fact_701_inf_OboundedE,axiom,
! [A: product_unit,B3: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B3 @ C ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ A @ B3 )
=> ~ ( ord_le3221252021190050221t_unit @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_702_inf_OboundedE,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B3 @ C ) )
=> ~ ( ( ord_less_eq_set_v @ A @ B3 )
=> ~ ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_703_inf_OboundedE,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B3 @ C ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_704_inf_OboundedI,axiom,
! [A: product_unit,B3: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B3 )
=> ( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_705_inf_OboundedI,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_706_inf_OboundedI,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_707_inf__greatest,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ Y4 )
=> ( ( ord_le3221252021190050221t_unit @ X3 @ Z2 )
=> ( ord_le3221252021190050221t_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_708_inf__greatest,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y4 )
=> ( ( ord_less_eq_set_v @ X3 @ Z2 )
=> ( ord_less_eq_set_v @ X3 @ ( inf_inf_set_v @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_709_inf__greatest,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y4 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Z2 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_710_inf_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A4: product_unit,B5: product_unit] :
( A4
= ( inf_inf_Product_unit @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_711_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B5: set_v] :
( A4
= ( inf_inf_set_v @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_712_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( A4
= ( inf_in6271465464967711157od_v_v @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_713_inf_Ocobounded1,axiom,
! [A: product_unit,B3: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B3 ) @ A ) ).
% inf.cobounded1
thf(fact_714_inf_Ocobounded1,axiom,
! [A: set_v,B3: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ A ) ).
% inf.cobounded1
thf(fact_715_inf_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ A ) ).
% inf.cobounded1
thf(fact_716_inf_Ocobounded2,axiom,
! [A: product_unit,B3: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_717_inf_Ocobounded2,axiom,
! [A: set_v,B3: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_718_inf_Ocobounded2,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_719_inf_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A4: product_unit,B5: product_unit] :
( ( inf_inf_Product_unit @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_720_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B5: set_v] :
( ( inf_inf_set_v @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_721_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_722_inf_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B5: product_unit,A4: product_unit] :
( ( inf_inf_Product_unit @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_723_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A4: set_v] :
( ( inf_inf_set_v @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_724_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_725_inf_OcoboundedI1,axiom,
! [A: product_unit,C: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_726_inf_OcoboundedI1,axiom,
! [A: set_v,C: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_727_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_728_inf_OcoboundedI2,axiom,
! [B3: product_unit,C: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B3 @ C )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_729_inf_OcoboundedI2,axiom,
! [B3: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B3 @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_730_inf_OcoboundedI2,axiom,
! [B3: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_731_inf_Oleft__commute,axiom,
! [B3: set_v,A: set_v,C: set_v] :
( ( inf_inf_set_v @ B3 @ ( inf_inf_set_v @ A @ C ) )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B3 @ C ) ) ) ).
% inf.left_commute
thf(fact_732_inf_Oleft__commute,axiom,
! [B3: product_unit,A: product_unit,C: product_unit] :
( ( inf_inf_Product_unit @ B3 @ ( inf_inf_Product_unit @ A @ C ) )
= ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ B3 @ C ) ) ) ).
% inf.left_commute
thf(fact_733_inf__left__commute,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( inf_inf_set_v @ X3 @ ( inf_inf_set_v @ Y4 @ Z2 ) )
= ( inf_inf_set_v @ Y4 @ ( inf_inf_set_v @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_734_inf__left__commute,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) )
= ( inf_inf_Product_unit @ Y4 @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_735_IntE,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A2 )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% IntE
thf(fact_736_IntE,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B ) )
=> ~ ( ( member_v @ C @ A2 )
=> ~ ( member_v @ C @ B ) ) ) ).
% IntE
thf(fact_737_IntD1,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A2 ) ) ).
% IntD1
thf(fact_738_IntD1,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B ) )
=> ( member_v @ C @ A2 ) ) ).
% IntD1
thf(fact_739_IntD2,axiom,
! [C: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ).
% IntD2
thf(fact_740_IntD2,axiom,
! [C: v,A2: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A2 @ B ) )
=> ( member_v @ C @ B ) ) ).
% IntD2
thf(fact_741_in__mono,axiom,
! [A2: set_v,B: set_v,X3: v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( member_v @ X3 @ A2 )
=> ( member_v @ X3 @ B ) ) ) ).
% in_mono
thf(fact_742_in__mono,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,X3: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( member7453568604450474000od_v_v @ X3 @ B ) ) ) ).
% in_mono
thf(fact_743_subsetD,axiom,
! [A2: set_v,B: set_v,C: v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( member_v @ C @ A2 )
=> ( member_v @ C @ B ) ) ) ).
% subsetD
thf(fact_744_subsetD,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( member7453568604450474000od_v_v @ C @ A2 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% subsetD
thf(fact_745_Int__mono,axiom,
! [A2: set_v,C2: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B ) @ ( inf_inf_set_v @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_746_Int__mono,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_747_Int__assoc,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A2 @ B ) @ C2 )
= ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_748_equalityE,axiom,
! [A2: set_v,B: set_v] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_v @ A2 @ B )
=> ~ ( ord_less_eq_set_v @ B @ A2 ) ) ) ).
% equalityE
thf(fact_749_equalityE,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A2 = B )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ A2 ) ) ) ).
% equalityE
thf(fact_750_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B2: set_v] :
! [X: v] :
( ( member_v @ X @ A3 )
=> ( member_v @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_751_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_752_Int__absorb,axiom,
! [A2: set_v] :
( ( inf_inf_set_v @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_753_Int__lower1,axiom,
! [A2: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_754_Int__lower1,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_755_Int__lower2,axiom,
! [A2: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_756_Int__lower2,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_757_equalityD1,axiom,
! [A2: set_v,B: set_v] :
( ( A2 = B )
=> ( ord_less_eq_set_v @ A2 @ B ) ) ).
% equalityD1
thf(fact_758_equalityD1,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A2 = B )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B ) ) ).
% equalityD1
thf(fact_759_equalityD2,axiom,
! [A2: set_v,B: set_v] :
( ( A2 = B )
=> ( ord_less_eq_set_v @ B @ A2 ) ) ).
% equalityD2
thf(fact_760_equalityD2,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A2 = B )
=> ( ord_le7336532860387713383od_v_v @ B @ A2 ) ) ).
% equalityD2
thf(fact_761_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B2: set_v] :
! [T2: v] :
( ( member_v @ T2 @ A3 )
=> ( member_v @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_762_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
! [T2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T2 @ A3 )
=> ( member7453568604450474000od_v_v @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_763_Int__absorb1,axiom,
! [B: set_v,A2: set_v] :
( ( ord_less_eq_set_v @ B @ A2 )
=> ( ( inf_inf_set_v @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_764_Int__absorb1,axiom,
! [B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_765_Int__absorb2,axiom,
! [A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( inf_inf_set_v @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_766_Int__absorb2,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_767_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A3: set_v,B2: set_v] : ( inf_inf_set_v @ B2 @ A3 ) ) ) ).
% Int_commute
thf(fact_768_subset__refl,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ A2 @ A2 ) ).
% subset_refl
thf(fact_769_subset__refl,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A2 @ A2 ) ).
% subset_refl
thf(fact_770_Collect__mono,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X2: set_v] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_mono
thf(fact_771_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X2: v] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_772_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X2: product_prod_v_v] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_773_Int__greatest,axiom,
! [C2: set_v,A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ A2 )
=> ( ( ord_less_eq_set_v @ C2 @ B )
=> ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_774_Int__greatest,axiom,
! [C2: set_Product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_775_subset__trans,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_776_subset__trans,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_777_set__eq__subset,axiom,
( ( ^ [Y2: set_v,Z: set_v] : ( Y2 = Z ) )
= ( ^ [A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
& ( ord_less_eq_set_v @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_778_set__eq__subset,axiom,
( ( ^ [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y2 = Z ) )
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
& ( ord_le7336532860387713383od_v_v @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_779_Int__left__absorb,axiom,
! [A2: set_v,B: set_v] :
( ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ A2 @ B ) )
= ( inf_inf_set_v @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_780_Collect__mono__iff,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) )
= ( ! [X: set_v] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_781_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X: v] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_782_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 ) )
= ( ! [X: product_prod_v_v] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_783_Int__Collect__mono,axiom,
! [A2: set_set_v,B: set_set_v,P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A2 @ B )
=> ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( inf_inf_set_set_v @ A2 @ ( collect_set_v @ P ) ) @ ( inf_inf_set_set_v @ B @ ( collect_set_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_784_Int__Collect__mono,axiom,
! [A2: set_v,B: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ! [X2: v] :
( ( member_v @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_785_Int__Collect__mono,axiom,
! [A2: 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 @ A2 @ B )
=> ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_786_Int__left__commute,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_787_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_v,Z: set_v] : ( Y2 = Z ) )
= ( ^ [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
& ( ord_less_eq_set_v @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_788_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y2 = Z ) )
= ( ^ [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
& ( ord_le7336532860387713383od_v_v @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_789_ord__eq__le__trans,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( A = B3 )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_790_ord__eq__le__trans,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A = B3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_791_ord__le__eq__trans,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_792_ord__le__eq__trans,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_793_order__antisym,axiom,
! [X3: set_v,Y4: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y4 )
=> ( ( ord_less_eq_set_v @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_794_order__antisym,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y4 )
=> ( ( ord_le7336532860387713383od_v_v @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_795_order_Otrans,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% order.trans
thf(fact_796_order_Otrans,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% order.trans
thf(fact_797_order__trans,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y4 )
=> ( ( ord_less_eq_set_v @ Y4 @ Z2 )
=> ( ord_less_eq_set_v @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_798_order__trans,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y4 )
=> ( ( ord_le7336532860387713383od_v_v @ Y4 @ Z2 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_799_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_v,Z: set_v] : ( Y2 = Z ) )
= ( ^ [A4: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ B5 @ A4 )
& ( ord_less_eq_set_v @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_800_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y2 = Z ) )
= ( ^ [A4: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B5 @ A4 )
& ( ord_le7336532860387713383od_v_v @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_801_dual__order_Oantisym,axiom,
! [B3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B3 @ A )
=> ( ( ord_less_eq_set_v @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_802_dual__order_Oantisym,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_803_dual__order_Otrans,axiom,
! [B3: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B3 @ A )
=> ( ( ord_less_eq_set_v @ C @ B3 )
=> ( ord_less_eq_set_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_804_dual__order_Otrans,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B3 )
=> ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_805_antisym,axiom,
! [A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ord_less_eq_set_v @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_806_antisym,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_807_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_v,Z: set_v] : ( Y2 = Z ) )
= ( ^ [A4: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ A4 @ B5 )
& ( ord_less_eq_set_v @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_808_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y2 = Z ) )
= ( ^ [A4: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B5 )
& ( ord_le7336532860387713383od_v_v @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_809_order__subst1,axiom,
! [A: set_v,F: set_v > set_v,B3: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ! [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_810_order__subst1,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_811_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B3: set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ! [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_812_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_813_order__subst2,axiom,
! [A: set_v,B3: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ord_less_eq_set_v @ ( F @ B3 ) @ C )
=> ( ! [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_814_order__subst2,axiom,
! [A: set_v,B3: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B3 ) @ C )
=> ( ! [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_815_order__subst2,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ord_less_eq_set_v @ ( F @ B3 ) @ C )
=> ( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_816_order__subst2,axiom,
! [A: set_Product_prod_v_v,B3: 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 @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B3 ) @ C )
=> ( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_817_order__eq__refl,axiom,
! [X3: set_v,Y4: set_v] :
( ( X3 = Y4 )
=> ( ord_less_eq_set_v @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_818_order__eq__refl,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( X3 = Y4 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_819_ord__eq__le__subst,axiom,
! [A: set_v,F: set_v > set_v,B3: set_v,C: set_v] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ! [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_820_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B3: set_v,C: set_v] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ! [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_821_ord__eq__le__subst,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_822_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_823_ord__le__eq__subst,axiom,
! [A: set_v,B3: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_824_ord__le__eq__subst,axiom,
! [A: set_v,B3: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_825_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_826_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B3: 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 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_827_order__antisym__conv,axiom,
! [Y4: set_v,X3: set_v] :
( ( ord_less_eq_set_v @ Y4 @ X3 )
=> ( ( ord_less_eq_set_v @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_828_order__antisym__conv,axiom,
! [Y4: set_Product_prod_v_v,X3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y4 @ X3 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_829_Collect__conj__eq,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( collect_set_v
@ ^ [X: set_v] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_830_Collect__conj__eq,axiom,
! [P: v > $o,Q: v > $o] :
( ( collect_v
@ ^ [X: v] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_831_Int__Collect,axiom,
! [X3: product_prod_v_v,A2: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ A2 @ ( collec140062887454715474od_v_v @ P ) ) )
= ( ( member7453568604450474000od_v_v @ X3 @ A2 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_832_Int__Collect,axiom,
! [X3: set_v,A2: set_set_v,P: set_v > $o] :
( ( member_set_v @ X3 @ ( inf_inf_set_set_v @ A2 @ ( collect_set_v @ P ) ) )
= ( ( member_set_v @ X3 @ A2 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_833_Int__Collect,axiom,
! [X3: v,A2: set_v,P: v > $o] :
( ( member_v @ X3 @ ( inf_inf_set_v @ A2 @ ( collect_v @ P ) ) )
= ( ( member_v @ X3 @ A2 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_834_Int__def,axiom,
( inf_in6271465464967711157od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
& ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_835_Int__def,axiom,
( inf_inf_set_set_v
= ( ^ [A3: set_set_v,B2: set_set_v] :
( collect_set_v
@ ^ [X: set_v] :
( ( member_set_v @ X @ A3 )
& ( member_set_v @ X @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_836_Int__def,axiom,
( inf_inf_set_v
= ( ^ [A3: set_v,B2: set_v] :
( collect_v
@ ^ [X: v] :
( ( member_v @ X @ A3 )
& ( member_v @ X @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_837_Collect__subset,axiom,
! [A2: set_set_v,P: set_v > $o] :
( ord_le5216385588623774835_set_v
@ ( collect_set_v
@ ^ [X: set_v] :
( ( member_set_v @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_838_Collect__subset,axiom,
! [A2: set_v,P: v > $o] :
( ord_less_eq_set_v
@ ( collect_v
@ ^ [X: v] :
( ( member_v @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_839_Collect__subset,axiom,
! [A2: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ord_le7336532860387713383od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_840_sup__Un__eq,axiom,
! [R3: set_v,S: set_v] :
( ( sup_sup_v_o
@ ^ [X: v] : ( member_v @ X @ R3 )
@ ^ [X: v] : ( member_v @ X @ S ) )
= ( ^ [X: v] : ( member_v @ X @ ( sup_sup_set_v @ R3 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_841_sup__Un__eq,axiom,
! [R3: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sup_su5941406310530359554_v_v_o
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ R3 )
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ S ) )
= ( ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ ( sup_su414716646722978715od_v_v @ R3 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_842_sup__Un__eq,axiom,
! [R3: set_set_v,S: set_set_v] :
( ( sup_sup_set_v_o
@ ^ [X: set_v] : ( member_set_v @ X @ R3 )
@ ^ [X: set_v] : ( member_set_v @ X @ S ) )
= ( ^ [X: set_v] : ( member_set_v @ X @ ( sup_sup_set_set_v @ R3 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_843_distrib__imp1,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ! [X2: set_v,Y3: set_v,Z4: set_v] :
( ( inf_inf_set_v @ X2 @ ( sup_sup_set_v @ Y3 @ Z4 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X2 @ Y3 ) @ ( inf_inf_set_v @ X2 @ Z4 ) ) )
=> ( ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ Y4 @ Z2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X3 @ Y4 ) @ ( sup_sup_set_v @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_844_distrib__imp1,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v,Z4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ Y3 @ Z4 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X2 @ Y3 ) @ ( inf_in6271465464967711157od_v_v @ X2 @ Z4 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y4 @ Z2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) @ ( sup_su414716646722978715od_v_v @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_845_distrib__imp1,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] :
( ! [X2: set_set_v,Y3: set_set_v,Z4: set_set_v] :
( ( inf_inf_set_set_v @ X2 @ ( sup_sup_set_set_v @ Y3 @ Z4 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X2 @ Y3 ) @ ( inf_inf_set_set_v @ X2 @ Z4 ) ) )
=> ( ( sup_sup_set_set_v @ X3 @ ( inf_inf_set_set_v @ Y4 @ Z2 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X3 @ Y4 ) @ ( sup_sup_set_set_v @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_846_distrib__imp1,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ! [X2: product_unit,Y3: product_unit,Z4: product_unit] :
( ( inf_inf_Product_unit @ X2 @ ( sup_sup_Product_unit @ Y3 @ Z4 ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X2 @ Y3 ) @ ( inf_inf_Product_unit @ X2 @ Z4 ) ) )
=> ( ( sup_sup_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_847_distrib__imp2,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ! [X2: set_v,Y3: set_v,Z4: set_v] :
( ( sup_sup_set_v @ X2 @ ( inf_inf_set_v @ Y3 @ Z4 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X2 @ Y3 ) @ ( sup_sup_set_v @ X2 @ Z4 ) ) )
=> ( ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ Y4 @ Z2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X3 @ Y4 ) @ ( inf_inf_set_v @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_848_distrib__imp2,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v,Z4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ ( inf_in6271465464967711157od_v_v @ Y3 @ Z4 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X2 @ Y3 ) @ ( sup_su414716646722978715od_v_v @ X2 @ Z4 ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y4 @ Z2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y4 ) @ ( inf_in6271465464967711157od_v_v @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_849_distrib__imp2,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] :
( ! [X2: set_set_v,Y3: set_set_v,Z4: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ ( inf_inf_set_set_v @ Y3 @ Z4 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X2 @ Y3 ) @ ( sup_sup_set_set_v @ X2 @ Z4 ) ) )
=> ( ( inf_inf_set_set_v @ X3 @ ( sup_sup_set_set_v @ Y4 @ Z2 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X3 @ Y4 ) @ ( inf_inf_set_set_v @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_850_distrib__imp2,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ! [X2: product_unit,Y3: product_unit,Z4: product_unit] :
( ( sup_sup_Product_unit @ X2 @ ( inf_inf_Product_unit @ Y3 @ Z4 ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X2 @ Y3 ) @ ( sup_sup_Product_unit @ X2 @ Z4 ) ) )
=> ( ( inf_inf_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_851_inf__sup__distrib1,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ Y4 @ Z2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X3 @ Y4 ) @ ( inf_inf_set_v @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_852_inf__sup__distrib1,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y4 @ Z2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y4 ) @ ( inf_in6271465464967711157od_v_v @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_853_inf__sup__distrib1,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] :
( ( inf_inf_set_set_v @ X3 @ ( sup_sup_set_set_v @ Y4 @ Z2 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X3 @ Y4 ) @ ( inf_inf_set_set_v @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_854_inf__sup__distrib1,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_855_inf__sup__distrib2,axiom,
! [Y4: set_v,Z2: set_v,X3: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y4 @ Z2 ) @ X3 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y4 @ X3 ) @ ( inf_inf_set_v @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_856_inf__sup__distrib2,axiom,
! [Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v,X3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y4 @ Z2 ) @ X3 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y4 @ X3 ) @ ( inf_in6271465464967711157od_v_v @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_857_inf__sup__distrib2,axiom,
! [Y4: set_set_v,Z2: set_set_v,X3: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y4 @ Z2 ) @ X3 )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y4 @ X3 ) @ ( inf_inf_set_set_v @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_858_inf__sup__distrib2,axiom,
! [Y4: product_unit,Z2: product_unit,X3: product_unit] :
( ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y4 @ Z2 ) @ X3 )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y4 @ X3 ) @ ( inf_inf_Product_unit @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_859_sup__inf__distrib1,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ Y4 @ Z2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X3 @ Y4 ) @ ( sup_sup_set_v @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_860_sup__inf__distrib1,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y4 @ Z2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) @ ( sup_su414716646722978715od_v_v @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_861_sup__inf__distrib1,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ ( inf_inf_set_set_v @ Y4 @ Z2 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X3 @ Y4 ) @ ( sup_sup_set_set_v @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_862_sup__inf__distrib1,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_863_sup__inf__distrib2,axiom,
! [Y4: set_v,Z2: set_v,X3: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y4 @ Z2 ) @ X3 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y4 @ X3 ) @ ( sup_sup_set_v @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_864_sup__inf__distrib2,axiom,
! [Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v,X3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y4 @ Z2 ) @ X3 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y4 @ X3 ) @ ( sup_su414716646722978715od_v_v @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_865_sup__inf__distrib2,axiom,
! [Y4: set_set_v,Z2: set_set_v,X3: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y4 @ Z2 ) @ X3 )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y4 @ X3 ) @ ( sup_sup_set_set_v @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_866_sup__inf__distrib2,axiom,
! [Y4: product_unit,Z2: product_unit,X3: product_unit] :
( ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y4 @ Z2 ) @ X3 )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y4 @ X3 ) @ ( sup_sup_Product_unit @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_867_disjoint__iff__not__equal,axiom,
! [A2: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A2 @ B )
= bot_bot_set_v )
= ( ! [X: v] :
( ( member_v @ X @ A2 )
=> ! [Y: v] :
( ( member_v @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_868_disjoint__iff__not__equal,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A2 )
=> ! [Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_869_disjoint__iff__not__equal,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A2 @ B )
= bot_bot_set_set_v )
= ( ! [X: set_v] :
( ( member_set_v @ X @ A2 )
=> ! [Y: set_v] :
( ( member_set_v @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_870_Int__empty__right,axiom,
! [A2: set_v] :
( ( inf_inf_set_v @ A2 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_871_Int__empty__right,axiom,
! [A2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_872_Int__empty__right,axiom,
! [A2: set_set_v] :
( ( inf_inf_set_set_v @ A2 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% Int_empty_right
thf(fact_873_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_874_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_875_Int__empty__left,axiom,
! [B: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ B )
= bot_bot_set_set_v ) ).
% Int_empty_left
thf(fact_876_disjoint__iff,axiom,
! [A2: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A2 @ B )
= bot_bot_set_v )
= ( ! [X: v] :
( ( member_v @ X @ A2 )
=> ~ ( member_v @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_877_disjoint__iff,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A2 )
=> ~ ( member7453568604450474000od_v_v @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_878_disjoint__iff,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A2 @ B )
= bot_bot_set_set_v )
= ( ! [X: set_v] :
( ( member_set_v @ X @ A2 )
=> ~ ( member_set_v @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_879_Int__emptyI,axiom,
! [A2: set_v,B: set_v] :
( ! [X2: v] :
( ( member_v @ X2 @ A2 )
=> ~ ( member_v @ X2 @ B ) )
=> ( ( inf_inf_set_v @ A2 @ B )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_880_Int__emptyI,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
=> ~ ( member7453568604450474000od_v_v @ X2 @ B ) )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_881_Int__emptyI,axiom,
! [A2: set_set_v,B: set_set_v] :
( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ~ ( member_set_v @ X2 @ B ) )
=> ( ( inf_inf_set_set_v @ A2 @ B )
= bot_bot_set_set_v ) ) ).
% Int_emptyI
thf(fact_882_Int__insert__left,axiom,
! [A: set_v,C2: set_set_v,B: set_set_v] :
( ( ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C2 )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ B @ C2 ) ) ) )
& ( ~ ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C2 )
= ( inf_inf_set_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_883_Int__insert__left,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_884_Int__insert__left,axiom,
! [A: v,C2: set_v,B: set_v] :
( ( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C2 )
= ( insert_v @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) )
& ( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C2 )
= ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_885_Int__insert__right,axiom,
! [A: set_v,A2: set_set_v,B: set_set_v] :
( ( ( member_set_v @ A @ A2 )
=> ( ( inf_inf_set_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A2 @ B ) ) ) )
& ( ~ ( member_set_v @ A @ A2 )
=> ( ( inf_inf_set_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( inf_inf_set_set_v @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_886_Int__insert__right,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_887_Int__insert__right,axiom,
! [A: v,A2: set_v,B: set_v] :
( ( ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A2 @ B ) ) ) )
& ( ~ ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( inf_inf_set_v @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_888_Un__Int__crazy,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B ) @ ( inf_inf_set_v @ B @ C2 ) ) @ ( inf_inf_set_v @ C2 @ A2 ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A2 @ B ) @ ( sup_sup_set_v @ B @ C2 ) ) @ ( sup_sup_set_v @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_889_Un__Int__crazy,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A2 ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) @ ( sup_su414716646722978715od_v_v @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_890_Un__Int__crazy,axiom,
! [A2: set_set_v,B: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A2 @ B ) @ ( inf_inf_set_set_v @ B @ C2 ) ) @ ( inf_inf_set_set_v @ C2 @ A2 ) )
= ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ ( sup_sup_set_set_v @ B @ C2 ) ) @ ( sup_sup_set_set_v @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_891_Int__Un__distrib,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ A2 @ ( sup_sup_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B ) @ ( inf_inf_set_v @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_892_Int__Un__distrib,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ ( inf_in6271465464967711157od_v_v @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_893_Int__Un__distrib,axiom,
! [A2: set_set_v,B: set_set_v,C2: set_set_v] :
( ( inf_inf_set_set_v @ A2 @ ( sup_sup_set_set_v @ B @ C2 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A2 @ B ) @ ( inf_inf_set_set_v @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_894_Un__Int__distrib,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ A2 @ ( inf_inf_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A2 @ B ) @ ( sup_sup_set_v @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_895_Un__Int__distrib,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ ( sup_su414716646722978715od_v_v @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_896_Un__Int__distrib,axiom,
! [A2: set_set_v,B: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( inf_inf_set_set_v @ B @ C2 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ ( sup_sup_set_set_v @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_897_Int__Un__distrib2,axiom,
! [B: set_v,C2: set_v,A2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B @ C2 ) @ A2 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B @ A2 ) @ ( inf_inf_set_v @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_898_Int__Un__distrib2,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C2 ) @ A2 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ A2 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_899_Int__Un__distrib2,axiom,
! [B: set_set_v,C2: set_set_v,A2: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B @ C2 ) @ A2 )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B @ A2 ) @ ( inf_inf_set_set_v @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_900_Un__Int__distrib2,axiom,
! [B: set_v,C2: set_v,A2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B @ C2 ) @ A2 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B @ A2 ) @ ( sup_sup_set_v @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_901_Un__Int__distrib2,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) @ A2 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ A2 ) @ ( sup_su414716646722978715od_v_v @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_902_Un__Int__distrib2,axiom,
! [B: set_set_v,C2: set_set_v,A2: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B @ C2 ) @ A2 )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B @ A2 ) @ ( sup_sup_set_set_v @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_903_Int__Diff,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ B ) @ C2 )
= ( inf_inf_set_v @ A2 @ ( minus_minus_set_v @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_904_Diff__Int2,axiom,
! [A2: set_v,C2: set_v,B: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_905_Diff__Diff__Int,axiom,
! [A2: set_v,B: set_v] :
( ( minus_minus_set_v @ A2 @ ( minus_minus_set_v @ A2 @ B ) )
= ( inf_inf_set_v @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_906_Diff__Int__distrib,axiom,
! [C2: set_v,A2: set_v,B: set_v] :
( ( inf_inf_set_v @ C2 @ ( minus_minus_set_v @ A2 @ B ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C2 @ A2 ) @ ( inf_inf_set_v @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_907_Diff__Int__distrib2,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A2 @ B ) @ C2 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_908_bot_Oextremum__uniqueI,axiom,
! [A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v )
=> ( A = bot_bot_set_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_909_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_910_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_911_bot_Oextremum__unique,axiom,
! [A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v )
= ( A = bot_bot_set_set_v ) ) ).
% bot.extremum_unique
thf(fact_912_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_913_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_914_bot_Oextremum,axiom,
! [A: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A ) ).
% bot.extremum
thf(fact_915_bot_Oextremum,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% bot.extremum
thf(fact_916_bot_Oextremum,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% bot.extremum
thf(fact_917_inf__sup__ord_I4_J,axiom,
! [Y4: set_set_v,X3: set_set_v] : ( ord_le5216385588623774835_set_v @ Y4 @ ( sup_sup_set_set_v @ X3 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_918_inf__sup__ord_I4_J,axiom,
! [Y4: product_unit,X3: product_unit] : ( ord_le3221252021190050221t_unit @ Y4 @ ( sup_sup_Product_unit @ X3 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_919_inf__sup__ord_I4_J,axiom,
! [Y4: set_v,X3: set_v] : ( ord_less_eq_set_v @ Y4 @ ( sup_sup_set_v @ X3 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_920_inf__sup__ord_I4_J,axiom,
! [Y4: set_Product_prod_v_v,X3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y4 @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_921_inf__sup__ord_I3_J,axiom,
! [X3: set_set_v,Y4: set_set_v] : ( ord_le5216385588623774835_set_v @ X3 @ ( sup_sup_set_set_v @ X3 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_922_inf__sup__ord_I3_J,axiom,
! [X3: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ X3 @ ( sup_sup_Product_unit @ X3 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_923_inf__sup__ord_I3_J,axiom,
! [X3: set_v,Y4: set_v] : ( ord_less_eq_set_v @ X3 @ ( sup_sup_set_v @ X3 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_924_inf__sup__ord_I3_J,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_925_le__supE,axiom,
! [A: set_set_v,B3: set_set_v,X3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B3 ) @ X3 )
=> ~ ( ( ord_le5216385588623774835_set_v @ A @ X3 )
=> ~ ( ord_le5216385588623774835_set_v @ B3 @ X3 ) ) ) ).
% le_supE
thf(fact_926_le__supE,axiom,
! [A: product_unit,B3: product_unit,X3: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A @ B3 ) @ X3 )
=> ~ ( ( ord_le3221252021190050221t_unit @ A @ X3 )
=> ~ ( ord_le3221252021190050221t_unit @ B3 @ X3 ) ) ) ).
% le_supE
thf(fact_927_le__supE,axiom,
! [A: set_v,B3: set_v,X3: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B3 ) @ X3 )
=> ~ ( ( ord_less_eq_set_v @ A @ X3 )
=> ~ ( ord_less_eq_set_v @ B3 @ X3 ) ) ) ).
% le_supE
thf(fact_928_le__supE,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,X3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B3 ) @ X3 )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ X3 )
=> ~ ( ord_le7336532860387713383od_v_v @ B3 @ X3 ) ) ) ).
% le_supE
thf(fact_929_le__supI,axiom,
! [A: set_set_v,X3: set_set_v,B3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ X3 )
=> ( ( ord_le5216385588623774835_set_v @ B3 @ X3 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B3 ) @ X3 ) ) ) ).
% le_supI
thf(fact_930_le__supI,axiom,
! [A: product_unit,X3: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ X3 )
=> ( ( ord_le3221252021190050221t_unit @ B3 @ X3 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A @ B3 ) @ X3 ) ) ) ).
% le_supI
thf(fact_931_le__supI,axiom,
! [A: set_v,X3: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ X3 )
=> ( ( ord_less_eq_set_v @ B3 @ X3 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B3 ) @ X3 ) ) ) ).
% le_supI
thf(fact_932_le__supI,axiom,
! [A: set_Product_prod_v_v,X3: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B3 ) @ X3 ) ) ) ).
% le_supI
thf(fact_933_sup__ge1,axiom,
! [X3: set_set_v,Y4: set_set_v] : ( ord_le5216385588623774835_set_v @ X3 @ ( sup_sup_set_set_v @ X3 @ Y4 ) ) ).
% sup_ge1
thf(fact_934_sup__ge1,axiom,
! [X3: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ X3 @ ( sup_sup_Product_unit @ X3 @ Y4 ) ) ).
% sup_ge1
thf(fact_935_sup__ge1,axiom,
! [X3: set_v,Y4: set_v] : ( ord_less_eq_set_v @ X3 @ ( sup_sup_set_v @ X3 @ Y4 ) ) ).
% sup_ge1
thf(fact_936_sup__ge1,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) ) ).
% sup_ge1
thf(fact_937_sup__ge2,axiom,
! [Y4: set_set_v,X3: set_set_v] : ( ord_le5216385588623774835_set_v @ Y4 @ ( sup_sup_set_set_v @ X3 @ Y4 ) ) ).
% sup_ge2
thf(fact_938_sup__ge2,axiom,
! [Y4: product_unit,X3: product_unit] : ( ord_le3221252021190050221t_unit @ Y4 @ ( sup_sup_Product_unit @ X3 @ Y4 ) ) ).
% sup_ge2
thf(fact_939_sup__ge2,axiom,
! [Y4: set_v,X3: set_v] : ( ord_less_eq_set_v @ Y4 @ ( sup_sup_set_v @ X3 @ Y4 ) ) ).
% sup_ge2
thf(fact_940_sup__ge2,axiom,
! [Y4: set_Product_prod_v_v,X3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y4 @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) ) ).
% sup_ge2
thf(fact_941_le__supI1,axiom,
! [X3: set_set_v,A: set_set_v,B3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X3 @ A )
=> ( ord_le5216385588623774835_set_v @ X3 @ ( sup_sup_set_set_v @ A @ B3 ) ) ) ).
% le_supI1
thf(fact_942_le__supI1,axiom,
! [X3: product_unit,A: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ A )
=> ( ord_le3221252021190050221t_unit @ X3 @ ( sup_sup_Product_unit @ A @ B3 ) ) ) ).
% le_supI1
thf(fact_943_le__supI1,axiom,
! [X3: set_v,A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ X3 @ A )
=> ( ord_less_eq_set_v @ X3 @ ( sup_sup_set_v @ A @ B3 ) ) ) ).
% le_supI1
thf(fact_944_le__supI1,axiom,
! [X3: set_Product_prod_v_v,A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ A )
=> ( ord_le7336532860387713383od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% le_supI1
thf(fact_945_le__supI2,axiom,
! [X3: set_set_v,B3: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X3 @ B3 )
=> ( ord_le5216385588623774835_set_v @ X3 @ ( sup_sup_set_set_v @ A @ B3 ) ) ) ).
% le_supI2
thf(fact_946_le__supI2,axiom,
! [X3: product_unit,B3: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ B3 )
=> ( ord_le3221252021190050221t_unit @ X3 @ ( sup_sup_Product_unit @ A @ B3 ) ) ) ).
% le_supI2
thf(fact_947_le__supI2,axiom,
! [X3: set_v,B3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ X3 @ B3 )
=> ( ord_less_eq_set_v @ X3 @ ( sup_sup_set_v @ A @ B3 ) ) ) ).
% le_supI2
thf(fact_948_le__supI2,axiom,
! [X3: set_Product_prod_v_v,B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ B3 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% le_supI2
thf(fact_949_sup_Omono,axiom,
! [C: set_set_v,A: set_set_v,D: set_set_v,B3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ A )
=> ( ( ord_le5216385588623774835_set_v @ D @ B3 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ C @ D ) @ ( sup_sup_set_set_v @ A @ B3 ) ) ) ) ).
% sup.mono
thf(fact_950_sup_Omono,axiom,
! [C: product_unit,A: product_unit,D: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ A )
=> ( ( ord_le3221252021190050221t_unit @ D @ B3 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ C @ D ) @ ( sup_sup_Product_unit @ A @ B3 ) ) ) ) ).
% sup.mono
thf(fact_951_sup_Omono,axiom,
! [C: set_v,A: set_v,D: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ( ord_less_eq_set_v @ D @ B3 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D ) @ ( sup_sup_set_v @ A @ B3 ) ) ) ) ).
% sup.mono
thf(fact_952_sup_Omono,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,D: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B3 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D ) @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ) ).
% sup.mono
thf(fact_953_sup__mono,axiom,
! [A: set_set_v,C: set_set_v,B3: set_set_v,D: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ C )
=> ( ( ord_le5216385588623774835_set_v @ B3 @ D )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B3 ) @ ( sup_sup_set_set_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_954_sup__mono,axiom,
! [A: product_unit,C: product_unit,B3: product_unit,D: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ( ord_le3221252021190050221t_unit @ B3 @ D )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A @ B3 ) @ ( sup_sup_Product_unit @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_955_sup__mono,axiom,
! [A: set_v,C: set_v,B3: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B3 @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B3 ) @ ( sup_sup_set_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_956_sup__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B3: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B3 ) @ ( sup_su414716646722978715od_v_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_957_sup__least,axiom,
! [Y4: set_set_v,X3: set_set_v,Z2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y4 @ X3 )
=> ( ( ord_le5216385588623774835_set_v @ Z2 @ X3 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ Y4 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_958_sup__least,axiom,
! [Y4: product_unit,X3: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y4 @ X3 )
=> ( ( ord_le3221252021190050221t_unit @ Z2 @ X3 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ Y4 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_959_sup__least,axiom,
! [Y4: set_v,X3: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ Y4 @ X3 )
=> ( ( ord_less_eq_set_v @ Z2 @ X3 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y4 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_960_sup__least,axiom,
! [Y4: set_Product_prod_v_v,X3: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y4 @ X3 )
=> ( ( ord_le7336532860387713383od_v_v @ Z2 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y4 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_961_le__iff__sup,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [X: set_set_v,Y: set_set_v] :
( ( sup_sup_set_set_v @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_962_le__iff__sup,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X: product_unit,Y: product_unit] :
( ( sup_sup_Product_unit @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_963_le__iff__sup,axiom,
( ord_less_eq_set_v
= ( ^ [X: set_v,Y: set_v] :
( ( sup_sup_set_v @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_964_le__iff__sup,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_965_sup_OorderE,axiom,
! [B3: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B3 @ A )
=> ( A
= ( sup_sup_set_set_v @ A @ B3 ) ) ) ).
% sup.orderE
thf(fact_966_sup_OorderE,axiom,
! [B3: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B3 @ A )
=> ( A
= ( sup_sup_Product_unit @ A @ B3 ) ) ) ).
% sup.orderE
thf(fact_967_sup_OorderE,axiom,
! [B3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B3 @ A )
=> ( A
= ( sup_sup_set_v @ A @ B3 ) ) ) ).
% sup.orderE
thf(fact_968_sup_OorderE,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( A
= ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% sup.orderE
thf(fact_969_sup_OorderI,axiom,
! [A: set_set_v,B3: set_set_v] :
( ( A
= ( sup_sup_set_set_v @ A @ B3 ) )
=> ( ord_le5216385588623774835_set_v @ B3 @ A ) ) ).
% sup.orderI
thf(fact_970_sup_OorderI,axiom,
! [A: product_unit,B3: product_unit] :
( ( A
= ( sup_sup_Product_unit @ A @ B3 ) )
=> ( ord_le3221252021190050221t_unit @ B3 @ A ) ) ).
% sup.orderI
thf(fact_971_sup_OorderI,axiom,
! [A: set_v,B3: set_v] :
( ( A
= ( sup_sup_set_v @ A @ B3 ) )
=> ( ord_less_eq_set_v @ B3 @ A ) ) ).
% sup.orderI
thf(fact_972_sup_OorderI,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ A @ B3 ) )
=> ( ord_le7336532860387713383od_v_v @ B3 @ A ) ) ).
% sup.orderI
thf(fact_973_sup__unique,axiom,
! [F: set_set_v > set_set_v > set_set_v,X3: set_set_v,Y4: set_set_v] :
( ! [X2: set_set_v,Y3: set_set_v] : ( ord_le5216385588623774835_set_v @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_set_v,Y3: set_set_v] : ( ord_le5216385588623774835_set_v @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_set_v,Y3: set_set_v,Z4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y3 @ X2 )
=> ( ( ord_le5216385588623774835_set_v @ Z4 @ X2 )
=> ( ord_le5216385588623774835_set_v @ ( F @ Y3 @ Z4 ) @ X2 ) ) )
=> ( ( sup_sup_set_set_v @ X3 @ Y4 )
= ( F @ X3 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_974_sup__unique,axiom,
! [F: product_unit > product_unit > product_unit,X3: product_unit,Y4: product_unit] :
( ! [X2: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: product_unit,Y3: product_unit,Z4: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y3 @ X2 )
=> ( ( ord_le3221252021190050221t_unit @ Z4 @ X2 )
=> ( ord_le3221252021190050221t_unit @ ( F @ Y3 @ Z4 ) @ X2 ) ) )
=> ( ( sup_sup_Product_unit @ X3 @ Y4 )
= ( F @ X3 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_975_sup__unique,axiom,
! [F: set_v > set_v > set_v,X3: set_v,Y4: set_v] :
( ! [X2: set_v,Y3: set_v] : ( ord_less_eq_set_v @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_v,Y3: set_v] : ( ord_less_eq_set_v @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_v,Y3: set_v,Z4: set_v] :
( ( ord_less_eq_set_v @ Y3 @ X2 )
=> ( ( ord_less_eq_set_v @ Z4 @ X2 )
=> ( ord_less_eq_set_v @ ( F @ Y3 @ Z4 ) @ X2 ) ) )
=> ( ( sup_sup_set_v @ X3 @ Y4 )
= ( F @ X3 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_976_sup__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v,Z4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y3 @ X2 )
=> ( ( ord_le7336532860387713383od_v_v @ Z4 @ X2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ Y3 @ Z4 ) @ X2 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X3 @ Y4 )
= ( F @ X3 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_977_sup_Oabsorb1,axiom,
! [B3: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B3 @ A )
=> ( ( sup_sup_set_set_v @ A @ B3 )
= A ) ) ).
% sup.absorb1
thf(fact_978_sup_Oabsorb1,axiom,
! [B3: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B3 @ A )
=> ( ( sup_sup_Product_unit @ A @ B3 )
= A ) ) ).
% sup.absorb1
thf(fact_979_sup_Oabsorb1,axiom,
! [B3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B3 @ A )
=> ( ( sup_sup_set_v @ A @ B3 )
= A ) ) ).
% sup.absorb1
thf(fact_980_sup_Oabsorb1,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B3 )
= A ) ) ).
% sup.absorb1
thf(fact_981_sup_Oabsorb2,axiom,
! [A: set_set_v,B3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B3 )
=> ( ( sup_sup_set_set_v @ A @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_982_sup_Oabsorb2,axiom,
! [A: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B3 )
=> ( ( sup_sup_Product_unit @ A @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_983_sup_Oabsorb2,axiom,
! [A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( sup_sup_set_v @ A @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_984_sup_Oabsorb2,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( sup_su414716646722978715od_v_v @ A @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_985_sup__absorb1,axiom,
! [Y4: set_set_v,X3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y4 @ X3 )
=> ( ( sup_sup_set_set_v @ X3 @ Y4 )
= X3 ) ) ).
% sup_absorb1
thf(fact_986_sup__absorb1,axiom,
! [Y4: product_unit,X3: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y4 @ X3 )
=> ( ( sup_sup_Product_unit @ X3 @ Y4 )
= X3 ) ) ).
% sup_absorb1
thf(fact_987_sup__absorb1,axiom,
! [Y4: set_v,X3: set_v] :
( ( ord_less_eq_set_v @ Y4 @ X3 )
=> ( ( sup_sup_set_v @ X3 @ Y4 )
= X3 ) ) ).
% sup_absorb1
thf(fact_988_sup__absorb1,axiom,
! [Y4: set_Product_prod_v_v,X3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y4 @ X3 )
=> ( ( sup_su414716646722978715od_v_v @ X3 @ Y4 )
= X3 ) ) ).
% sup_absorb1
thf(fact_989_sup__absorb2,axiom,
! [X3: set_set_v,Y4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X3 @ Y4 )
=> ( ( sup_sup_set_set_v @ X3 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_990_sup__absorb2,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ Y4 )
=> ( ( sup_sup_Product_unit @ X3 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_991_sup__absorb2,axiom,
! [X3: set_v,Y4: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y4 )
=> ( ( sup_sup_set_v @ X3 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_992_sup__absorb2,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y4 )
=> ( ( sup_su414716646722978715od_v_v @ X3 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_993_sup_OboundedE,axiom,
! [B3: set_set_v,C: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B3 @ C ) @ A )
=> ~ ( ( ord_le5216385588623774835_set_v @ B3 @ A )
=> ~ ( ord_le5216385588623774835_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_994_sup_OboundedE,axiom,
! [B3: product_unit,C: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B3 @ C ) @ A )
=> ~ ( ( ord_le3221252021190050221t_unit @ B3 @ A )
=> ~ ( ord_le3221252021190050221t_unit @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_995_sup_OboundedE,axiom,
! [B3: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B3 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_v @ B3 @ A )
=> ~ ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_996_sup_OboundedE,axiom,
! [B3: 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 @ B3 @ C ) @ A )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_997_sup_OboundedI,axiom,
! [B3: set_set_v,A: set_set_v,C: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B3 @ A )
=> ( ( ord_le5216385588623774835_set_v @ C @ A )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B3 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_998_sup_OboundedI,axiom,
! [B3: product_unit,A: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ B3 @ A )
=> ( ( ord_le3221252021190050221t_unit @ C @ A )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B3 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_999_sup_OboundedI,axiom,
! [B3: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B3 @ A )
=> ( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B3 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1000_sup_OboundedI,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B3 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1001_sup_Oorder__iff,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [B5: set_set_v,A4: set_set_v] :
( A4
= ( sup_sup_set_set_v @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1002_sup_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B5: product_unit,A4: product_unit] :
( A4
= ( sup_sup_Product_unit @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1003_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A4: set_v] :
( A4
= ( sup_sup_set_v @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1004_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( A4
= ( sup_su414716646722978715od_v_v @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1005_sup_Ocobounded1,axiom,
! [A: set_set_v,B3: set_set_v] : ( ord_le5216385588623774835_set_v @ A @ ( sup_sup_set_set_v @ A @ B3 ) ) ).
% sup.cobounded1
thf(fact_1006_sup_Ocobounded1,axiom,
! [A: product_unit,B3: product_unit] : ( ord_le3221252021190050221t_unit @ A @ ( sup_sup_Product_unit @ A @ B3 ) ) ).
% sup.cobounded1
thf(fact_1007_sup_Ocobounded1,axiom,
! [A: set_v,B3: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B3 ) ) ).
% sup.cobounded1
thf(fact_1008_sup_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ).
% sup.cobounded1
thf(fact_1009_sup_Ocobounded2,axiom,
! [B3: set_set_v,A: set_set_v] : ( ord_le5216385588623774835_set_v @ B3 @ ( sup_sup_set_set_v @ A @ B3 ) ) ).
% sup.cobounded2
thf(fact_1010_sup_Ocobounded2,axiom,
! [B3: product_unit,A: product_unit] : ( ord_le3221252021190050221t_unit @ B3 @ ( sup_sup_Product_unit @ A @ B3 ) ) ).
% sup.cobounded2
thf(fact_1011_sup_Ocobounded2,axiom,
! [B3: set_v,A: set_v] : ( ord_less_eq_set_v @ B3 @ ( sup_sup_set_v @ A @ B3 ) ) ).
% sup.cobounded2
thf(fact_1012_sup_Ocobounded2,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B3 @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ).
% sup.cobounded2
thf(fact_1013_sup_Oabsorb__iff1,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [B5: set_set_v,A4: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1014_sup_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B5: product_unit,A4: product_unit] :
( ( sup_sup_Product_unit @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1015_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A4: set_v] :
( ( sup_sup_set_v @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1016_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1017_sup_Oabsorb__iff2,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A4: set_set_v,B5: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1018_sup_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A4: product_unit,B5: product_unit] :
( ( sup_sup_Product_unit @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1019_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B5: set_v] :
( ( sup_sup_set_v @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1020_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1021_sup_OcoboundedI1,axiom,
! [C: set_set_v,A: set_set_v,B3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ A )
=> ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_1022_sup_OcoboundedI1,axiom,
! [C: product_unit,A: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ A )
=> ( ord_le3221252021190050221t_unit @ C @ ( sup_sup_Product_unit @ A @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_1023_sup_OcoboundedI1,axiom,
! [C: set_v,A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_1024_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_1025_sup_OcoboundedI2,axiom,
! [C: set_set_v,B3: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ B3 )
=> ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_1026_sup_OcoboundedI2,axiom,
! [C: product_unit,B3: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ B3 )
=> ( ord_le3221252021190050221t_unit @ C @ ( sup_sup_Product_unit @ A @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_1027_sup_OcoboundedI2,axiom,
! [C: set_v,B3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ C @ B3 )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_1028_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_v_v,B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ B3 )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_1029_insert__mono,axiom,
! [C2: set_set_v,D2: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ D2 )
=> ( ord_le5216385588623774835_set_v @ ( insert_set_v @ A @ C2 ) @ ( insert_set_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1030_insert__mono,axiom,
! [C2: set_v,D2: set_v,A: v] :
( ( ord_less_eq_set_v @ C2 @ D2 )
=> ( ord_less_eq_set_v @ ( insert_v @ A @ C2 ) @ ( insert_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1031_insert__mono,axiom,
! [C2: set_Product_prod_v_v,D2: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C2 ) @ ( insert1338601472111419319od_v_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1032_subset__insert,axiom,
! [X3: set_v,A2: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X3 @ A2 )
=> ( ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ X3 @ B ) )
= ( ord_le5216385588623774835_set_v @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1033_subset__insert,axiom,
! [X3: v,A2: set_v,B: set_v] :
( ~ ( member_v @ X3 @ A2 )
=> ( ( ord_less_eq_set_v @ A2 @ ( insert_v @ X3 @ B ) )
= ( ord_less_eq_set_v @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1034_subset__insert,axiom,
! [X3: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X3 @ B ) )
= ( ord_le7336532860387713383od_v_v @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1035_subset__insertI,axiom,
! [B: set_set_v,A: set_v] : ( ord_le5216385588623774835_set_v @ B @ ( insert_set_v @ A @ B ) ) ).
% subset_insertI
thf(fact_1036_subset__insertI,axiom,
! [B: set_v,A: v] : ( ord_less_eq_set_v @ B @ ( insert_v @ A @ B ) ) ).
% subset_insertI
thf(fact_1037_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_1038_subset__insertI2,axiom,
! [A2: set_set_v,B: set_set_v,B3: set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B )
=> ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_1039_subset__insertI2,axiom,
! [A2: set_v,B: set_v,B3: v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ord_less_eq_set_v @ A2 @ ( insert_v @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_1040_subset__insertI2,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,B3: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_1041_Un__mono,axiom,
! [A2: set_set_v,C2: set_set_v,B: set_set_v,D2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ C2 )
=> ( ( ord_le5216385588623774835_set_v @ B @ D2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ ( sup_sup_set_set_v @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1042_Un__mono,axiom,
! [A2: set_v,C2: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B ) @ ( sup_sup_set_v @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1043_Un__mono,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ ( sup_su414716646722978715od_v_v @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1044_Un__least,axiom,
! [A2: set_set_v,C2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ C2 )
=> ( ( ord_le5216385588623774835_set_v @ B @ C2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_1045_Un__least,axiom,
! [A2: set_v,C2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_1046_Un__least,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_1047_Un__upper1,axiom,
! [A2: set_set_v,B: set_set_v] : ( ord_le5216385588623774835_set_v @ A2 @ ( sup_sup_set_set_v @ A2 @ B ) ) ).
% Un_upper1
thf(fact_1048_Un__upper1,axiom,
! [A2: set_v,B: set_v] : ( ord_less_eq_set_v @ A2 @ ( sup_sup_set_v @ A2 @ B ) ) ).
% Un_upper1
thf(fact_1049_Un__upper1,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ).
% Un_upper1
thf(fact_1050_Un__upper2,axiom,
! [B: set_set_v,A2: set_set_v] : ( ord_le5216385588623774835_set_v @ B @ ( sup_sup_set_set_v @ A2 @ B ) ) ).
% Un_upper2
thf(fact_1051_Un__upper2,axiom,
! [B: set_v,A2: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A2 @ B ) ) ).
% Un_upper2
thf(fact_1052_Un__upper2,axiom,
! [B: set_Product_prod_v_v,A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ).
% Un_upper2
thf(fact_1053_Un__absorb1,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B )
=> ( ( sup_sup_set_set_v @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1054_Un__absorb1,axiom,
! [A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( sup_sup_set_v @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1055_Un__absorb1,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1056_Un__absorb2,axiom,
! [B: set_set_v,A2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B @ A2 )
=> ( ( sup_sup_set_set_v @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_1057_Un__absorb2,axiom,
! [B: set_v,A2: set_v] :
( ( ord_less_eq_set_v @ B @ A2 )
=> ( ( sup_sup_set_v @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_1058_Un__absorb2,axiom,
! [B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A2 )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_1059_subset__UnE,axiom,
! [C2: set_set_v,A2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ ( sup_sup_set_set_v @ A2 @ B ) )
=> ~ ! [A5: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A5 @ A2 )
=> ! [B6: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B6 @ B )
=> ( C2
!= ( sup_sup_set_set_v @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1060_subset__UnE,axiom,
! [C2: set_v,A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A2 @ B ) )
=> ~ ! [A5: set_v] :
( ( ord_less_eq_set_v @ A5 @ A2 )
=> ! [B6: set_v] :
( ( ord_less_eq_set_v @ B6 @ B )
=> ( C2
!= ( sup_sup_set_v @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1061_subset__UnE,axiom,
! [C2: set_Product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A2 @ B ) )
=> ~ ! [A5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A5 @ A2 )
=> ! [B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B6 @ B )
=> ( C2
!= ( sup_su414716646722978715od_v_v @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1062_subset__Un__eq,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A3: set_set_v,B2: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_1063_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B2: set_v] :
( ( sup_sup_set_v @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_1064_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_1065_Diff__mono,axiom,
! [A2: set_v,C2: set_v,D2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ C2 )
=> ( ( ord_less_eq_set_v @ D2 @ B )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B ) @ ( minus_minus_set_v @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1066_Diff__mono,axiom,
! [A2: set_Product_prod_v_v,C2: set_Product_prod_v_v,D2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ ( minus_4183494784930505774od_v_v @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1067_Diff__subset,axiom,
! [A2: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_1068_Diff__subset,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_1069_double__diff,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1070_double__diff,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1071_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_1072_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_1073_sup__set__def,axiom,
( sup_sup_set_v
= ( ^ [A3: set_v,B2: set_v] :
( collect_v
@ ( sup_sup_v_o
@ ^ [X: v] : ( member_v @ X @ A3 )
@ ^ [X: v] : ( member_v @ X @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_1074_sup__set__def,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( sup_su5941406310530359554_v_v_o
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ A3 )
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_1075_sup__set__def,axiom,
( sup_sup_set_set_v
= ( ^ [A3: set_set_v,B2: set_set_v] :
( collect_set_v
@ ( sup_sup_set_v_o
@ ^ [X: set_v] : ( member_set_v @ X @ A3 )
@ ^ [X: set_v] : ( member_set_v @ X @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_1076_Diff__triv,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_1077_Diff__triv,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A2 @ B )
= bot_bot_set_set_v )
=> ( ( minus_7228012346218142266_set_v @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_1078_Diff__triv,axiom,
! [A2: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A2 @ B )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_1079_Int__Diff__disjoint,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ ( minus_4183494784930505774od_v_v @ A2 @ B ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_1080_Int__Diff__disjoint,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ A2 @ B ) @ ( minus_7228012346218142266_set_v @ A2 @ B ) )
= bot_bot_set_set_v ) ).
% Int_Diff_disjoint
thf(fact_1081_Int__Diff__disjoint,axiom,
! [A2: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A2 @ B ) @ ( minus_minus_set_v @ A2 @ B ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_1082_Un__Diff__Int,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_1083_Un__Diff__Int,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B ) @ ( inf_inf_set_set_v @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_1084_Un__Diff__Int,axiom,
! [A2: set_v,B: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ B ) @ ( inf_inf_set_v @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_1085_Int__Diff__Un,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ ( minus_4183494784930505774od_v_v @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_1086_Int__Diff__Un,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A2 @ B ) @ ( minus_7228012346218142266_set_v @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_1087_Int__Diff__Un,axiom,
! [A2: set_v,B: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B ) @ ( minus_minus_set_v @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_1088_Diff__Int,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ ( minus_4183494784930505774od_v_v @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1089_Diff__Int,axiom,
! [A2: set_set_v,B: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ ( inf_inf_set_set_v @ B @ C2 ) )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B ) @ ( minus_7228012346218142266_set_v @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1090_Diff__Int,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ A2 @ ( inf_inf_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ B ) @ ( minus_minus_set_v @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1091_Diff__Un,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ ( minus_4183494784930505774od_v_v @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1092_Diff__Un,axiom,
! [A2: set_set_v,B: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ ( sup_sup_set_set_v @ B @ C2 ) )
= ( inf_inf_set_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B ) @ ( minus_7228012346218142266_set_v @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1093_Diff__Un,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ A2 @ ( sup_sup_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A2 @ B ) @ ( minus_minus_set_v @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1094_subset__singleton__iff,axiom,
! [X7: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ X7 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
= ( ( X7 = bot_bot_set_set_v )
| ( X7
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_1095_subset__singleton__iff,axiom,
! [X7: set_v,A: v] :
( ( ord_less_eq_set_v @ X7 @ ( insert_v @ A @ bot_bot_set_v ) )
= ( ( X7 = bot_bot_set_v )
| ( X7
= ( insert_v @ A @ bot_bot_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_1096_subset__singleton__iff,axiom,
! [X7: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X7 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( ( X7 = bot_bo723834152578015283od_v_v )
| ( X7
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_1097_subset__singletonD,axiom,
! [A2: set_set_v,X3: set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ X3 @ bot_bot_set_set_v ) )
=> ( ( A2 = bot_bot_set_set_v )
| ( A2
= ( insert_set_v @ X3 @ bot_bot_set_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_1098_subset__singletonD,axiom,
! [A2: set_v,X3: v] :
( ( ord_less_eq_set_v @ A2 @ ( insert_v @ X3 @ bot_bot_set_v ) )
=> ( ( A2 = bot_bot_set_v )
| ( A2
= ( insert_v @ X3 @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_1099_subset__singletonD,axiom,
! [A2: set_Product_prod_v_v,X3: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
=> ( ( A2 = bot_bo723834152578015283od_v_v )
| ( A2
= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_1100_subset__Diff__insert,axiom,
! [A2: set_set_v,B: set_set_v,X3: set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B @ ( insert_set_v @ X3 @ C2 ) ) )
= ( ( ord_le5216385588623774835_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B @ C2 ) )
& ~ ( member_set_v @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1101_subset__Diff__insert,axiom,
! [A2: set_v,B: set_v,X3: v,C2: set_v] :
( ( ord_less_eq_set_v @ A2 @ ( minus_minus_set_v @ B @ ( insert_v @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_v @ A2 @ ( minus_minus_set_v @ B @ C2 ) )
& ~ ( member_v @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1102_subset__Diff__insert,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,X3: product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B @ ( insert1338601472111419319od_v_v @ X3 @ C2 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1103_Diff__subset__conv,axiom,
! [A2: set_set_v,B: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B ) @ C2 )
= ( ord_le5216385588623774835_set_v @ A2 @ ( sup_sup_set_set_v @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1104_Diff__subset__conv,axiom,
! [A2: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B ) @ C2 )
= ( ord_less_eq_set_v @ A2 @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1105_Diff__subset__conv,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ C2 )
= ( ord_le7336532860387713383od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1106_Diff__partition,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B )
=> ( ( sup_sup_set_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_1107_Diff__partition,axiom,
! [A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( sup_sup_set_v @ A2 @ ( minus_minus_set_v @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_1108_Diff__partition,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_1109_graph_Ora__mono,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y4 @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y4 @ E5 ) ) ) ) ).
% graph.ra_mono
thf(fact_1110_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,X3: 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 @ X3 @ ( inf_in6271465464967711157od_v_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_1111_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,S3: set_v,X3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( member_v @ X3 @ ( inf_inf_set_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_1112_graph_Odfs__S__tl__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) ) ) ).
% graph.dfs_S_tl_stack(2)
thf(fact_1113_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_1114_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_1115_graph_Ounite__subscc,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ 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 ) )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) @ ( hd_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_1116_graph_Ounite__subscc,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ 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 ) )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_1117_graph_Odfs__S__hd__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( 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 @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(2)
thf(fact_1118_graph_Odfs__S__hd__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(1)
thf(fact_1119_Diff__single__insert,axiom,
! [A2: set_set_v,X3: set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ X3 @ bot_bot_set_set_v ) ) @ B )
=> ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1120_Diff__single__insert,axiom,
! [A2: set_v,X3: v,B: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v @ X3 @ bot_bot_set_v ) ) @ B )
=> ( ord_less_eq_set_v @ A2 @ ( insert_v @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1121_Diff__single__insert,axiom,
! [A2: set_Product_prod_v_v,X3: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) @ B )
=> ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1122_subset__insert__iff,axiom,
! [A2: set_set_v,X3: set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ X3 @ B ) )
= ( ( ( member_set_v @ X3 @ A2 )
=> ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ X3 @ bot_bot_set_set_v ) ) @ B ) )
& ( ~ ( member_set_v @ X3 @ A2 )
=> ( ord_le5216385588623774835_set_v @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1123_subset__insert__iff,axiom,
! [A2: set_v,X3: v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ ( insert_v @ X3 @ B ) )
= ( ( ( member_v @ X3 @ A2 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v @ X3 @ bot_bot_set_v ) ) @ B ) )
& ( ~ ( member_v @ X3 @ A2 )
=> ( ord_less_eq_set_v @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1124_subset__insert__iff,axiom,
! [A2: set_Product_prod_v_v,X3: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X3 @ B ) )
= ( ( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) @ B ) )
& ( ~ ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1125_graph_Odfs__S__tl__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ).
% graph.dfs_S_tl_stack(1)
thf(fact_1126_graph_Ois__scc__def,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S: set_set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl1515522642333523865_set_v @ Successors @ S )
= ( ( S != bot_bot_set_set_v )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S )
& ! [S2: set_set_v] :
( ( ( ord_le5216385588623774835_set_v @ S @ S2 )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S2 ) )
=> ( S2 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_1127_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_1128_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_1129_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_1130_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_1131_set__empty2,axiom,
! [Xs: list_set_v] :
( ( bot_bot_set_set_v
= ( set_set_v2 @ Xs ) )
= ( Xs = nil_set_v ) ) ).
% set_empty2
thf(fact_1132_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_1133_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_1134_set__empty,axiom,
! [Xs: list_set_v] :
( ( ( set_set_v2 @ Xs )
= bot_bot_set_set_v )
= ( Xs = nil_set_v ) ) ).
% set_empty
thf(fact_1135_post__dfss__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl6082031138996704384t_unit @ successors @ V @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V )
= ( successors @ V ) )
& ! [X: v] :
( ( member_v @ X @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X )
= ( sCC_Bl3795065053823578884t_unit @ E @ X ) ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ! [X: v] :
( ( member_v @ X @ ( successors @ V ) )
=> ( member_v @ X @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E2 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) )
& ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ V ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X )
= ( sCC_Bl1280885523602775798t_unit @ E @ X ) ) )
& ( ( ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
= V )
=> ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ V @ X ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfss_def
thf(fact_1136_avoiding__explored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,X3: v,Y4: v,E4: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y4 @ E4 )
=> ( ~ ( member_v @ Y4 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y4 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% avoiding_explored
thf(fact_1137_boolean__algebra_Oconj__zero__right,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ X3 @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_right
thf(fact_1138_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_v] :
( ( inf_inf_set_v @ X3 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_1139_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_1140_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_set_v] :
( ( inf_inf_set_set_v @ X3 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_1141_boolean__algebra_Oconj__zero__left,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X3 )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_left
thf(fact_1142_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X3 )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_1143_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X3 )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_1144_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X3 )
= bot_bot_set_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_1145_post__dfs__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V )
= ( successors @ V ) )
& ! [X: v] :
( ( member_v @ X @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X )
= ( sCC_Bl3795065053823578884t_unit @ E @ X ) ) )
& ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( ( ( member_v @ V @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
= ( sCC_Bl8828226123343373779t_unit @ E ) )
& ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X )
= ( sCC_Bl1280885523602775798t_unit @ E @ X ) ) ) )
| ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X )
= ( sCC_Bl1280885523602775798t_unit @ E @ X ) ) ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfs_def
thf(fact_1146_ra__cases,axiom,
! [X3: v,Y4: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y4 @ E4 )
=> ( ( X3 = Y4 )
| ? [Z4: v] :
( ( member_v @ Z4 @ ( successors @ X3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Z4 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z4 @ Y4 @ E4 ) ) ) ) ).
% ra_cases
thf(fact_1147_edge__ra,axiom,
! [Y4: v,X3: v,E4: set_Product_prod_v_v] :
( ( member_v @ Y4 @ ( successors @ X3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y4 ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y4 @ E4 ) ) ) ).
% edge_ra
thf(fact_1148_reachable__avoiding_Osimps,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
= ( ? [X: v,E6: set_Product_prod_v_v] :
( ( A1 = X )
& ( A22 = X )
& ( A32 = E6 ) )
| ? [X: v,Y: v,E6: set_Product_prod_v_v,Z3: v] :
( ( A1 = X )
& ( A22 = Z3 )
& ( A32 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E6 )
& ( member_v @ Z3 @ ( successors @ Y ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z3 ) @ E6 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_1149_ra__succ,axiom,
! [X3: v,Y4: v,E4: set_Product_prod_v_v,Z2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y4 @ E4 )
=> ( ( member_v @ Z2 @ ( successors @ Y4 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y4 @ Z2 ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Z2 @ E4 ) ) ) ) ).
% ra_succ
thf(fact_1150_reachable__avoiding_Ocases,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y3 @ A32 )
=> ( ( member_v @ A22 @ ( successors @ Y3 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ A22 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_1151_ra__add__edge,axiom,
! [X3: v,Y4: v,E4: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y4 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y4 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ V @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y4 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_1152_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_1153_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_1154_set__append,axiom,
! [Xs: list_set_v,Ys: list_set_v] :
( ( set_set_v2 @ ( append_set_v @ Xs @ Ys ) )
= ( sup_sup_set_set_v @ ( set_set_v2 @ Xs ) @ ( set_set_v2 @ Ys ) ) ) ).
% set_append
thf(fact_1155_hd__append2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs != nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Xs ) ) ) ).
% hd_append2
thf(fact_1156_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_1157_sup__Un__eq2,axiom,
! [R3: set_Pr6425124735969554649t_unit,S: set_Pr6425124735969554649t_unit] :
( ( sup_su4631456023315819628unit_o
@ ^ [X: v,Y: sCC_Bl1394983891496994913t_unit] : ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X @ Y ) @ R3 )
@ ^ [X: v,Y: sCC_Bl1394983891496994913t_unit] : ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X @ Y ) @ S ) )
= ( ^ [X: v,Y: sCC_Bl1394983891496994913t_unit] : ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X @ Y ) @ ( sup_su7786205842889497605t_unit @ R3 @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_1158_sup__Un__eq2,axiom,
! [R3: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sup_sup_v_v_o
@ ^ [X: v,Y: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R3 )
@ ^ [X: v,Y: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ S ) )
= ( ^ [X: v,Y: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ R3 @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_1159_inf__Int__eq2,axiom,
! [R3: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( inf_inf_v_v_o
@ ^ [X: v,Y: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R3 )
@ ^ [X: v,Y: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ S ) )
= ( ^ [X: v,Y: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ R3 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_1160_inf__Int__eq2,axiom,
! [R3: set_Pr6425124735969554649t_unit,S: set_Pr6425124735969554649t_unit] :
( ( inf_in7082385440315676882unit_o
@ ^ [X: v,Y: sCC_Bl1394983891496994913t_unit] : ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X @ Y ) @ R3 )
@ ^ [X: v,Y: sCC_Bl1394983891496994913t_unit] : ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X @ Y ) @ S ) )
= ( ^ [X: v,Y: sCC_Bl1394983891496994913t_unit] : ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X @ Y ) @ ( inf_in5199935713740186731t_unit @ R3 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_1161_pred__equals__eq2,axiom,
! [R3: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( ( ^ [X: v,Y: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R3 ) )
= ( ^ [X: v,Y: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ S ) ) )
= ( R3 = S ) ) ).
% pred_equals_eq2
thf(fact_1162_pred__equals__eq2,axiom,
! [R3: set_Pr6425124735969554649t_unit,S: set_Pr6425124735969554649t_unit] :
( ( ( ^ [X: v,Y: sCC_Bl1394983891496994913t_unit] : ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X @ Y ) @ R3 ) )
= ( ^ [X: v,Y: sCC_Bl1394983891496994913t_unit] : ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X @ Y ) @ S ) ) )
= ( R3 = S ) ) ).
% pred_equals_eq2
thf(fact_1163_pred__subset__eq2,axiom,
! [R3: set_Pr6425124735969554649t_unit,S: set_Pr6425124735969554649t_unit] :
( ( ord_le7518007513909176736unit_o
@ ^ [X: v,Y: sCC_Bl1394983891496994913t_unit] : ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X @ Y ) @ R3 )
@ ^ [X: v,Y: sCC_Bl1394983891496994913t_unit] : ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X @ Y ) @ S ) )
= ( ord_le7290744839000465721t_unit @ R3 @ S ) ) ).
% pred_subset_eq2
thf(fact_1164_pred__subset__eq2,axiom,
! [R3: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( ord_less_eq_v_v_o
@ ^ [X: v,Y: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R3 )
@ ^ [X: v,Y: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ S ) )
= ( ord_le7336532860387713383od_v_v @ R3 @ S ) ) ).
% pred_subset_eq2
thf(fact_1165_subrelI,axiom,
! [R: set_Pr6425124735969554649t_unit,S4: set_Pr6425124735969554649t_unit] :
( ! [X2: v,Y3: sCC_Bl1394983891496994913t_unit] :
( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X2 @ Y3 ) @ R )
=> ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X2 @ Y3 ) @ S4 ) )
=> ( ord_le7290744839000465721t_unit @ R @ S4 ) ) ).
% subrelI
thf(fact_1166_subrelI,axiom,
! [R: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
( ! [X2: v,Y3: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ R )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ S4 ) )
=> ( ord_le7336532860387713383od_v_v @ R @ S4 ) ) ).
% subrelI
thf(fact_1167_hd__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( Xs = nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Ys ) ) )
& ( ( Xs != nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Xs ) ) ) ) ).
% hd_append
thf(fact_1168_longest__common__prefix,axiom,
! [Xs: list_v,Ys: list_v] :
? [Ps: list_v,Xs2: list_v,Ys2: list_v] :
( ( Xs
= ( append_v @ Ps @ Xs2 ) )
& ( Ys
= ( append_v @ Ps @ Ys2 ) )
& ( ( Xs2 = nil_v )
| ( Ys2 = nil_v )
| ( ( hd_v @ Xs2 )
!= ( hd_v @ Ys2 ) ) ) ) ).
% longest_common_prefix
thf(fact_1169_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_1170_inf__set__def,axiom,
( inf_in6271465464967711157od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( inf_in6860806757119575912_v_v_o
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ A3 )
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_1171_inf__set__def,axiom,
( inf_inf_set_set_v
= ( ^ [A3: set_set_v,B2: set_set_v] :
( collect_set_v
@ ( inf_inf_set_v_o
@ ^ [X: set_v] : ( member_set_v @ X @ A3 )
@ ^ [X: set_v] : ( member_set_v @ X @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_1172_inf__set__def,axiom,
( inf_inf_set_v
= ( ^ [A3: set_v,B2: set_v] :
( collect_v
@ ( inf_inf_v_o
@ ^ [X: v] : ( member_v @ X @ A3 )
@ ^ [X: v] : ( member_v @ X @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_1173_inf__Int__eq,axiom,
! [R3: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( inf_in6860806757119575912_v_v_o
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ R3 )
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ S ) )
= ( ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ R3 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1174_inf__Int__eq,axiom,
! [R3: set_v,S: set_v] :
( ( inf_inf_v_o
@ ^ [X: v] : ( member_v @ X @ R3 )
@ ^ [X: v] : ( member_v @ X @ S ) )
= ( ^ [X: v] : ( member_v @ X @ ( inf_inf_set_v @ R3 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1175_less__eq__set__def,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B2: set_v] :
( ord_less_eq_v_o
@ ^ [X: v] : ( member_v @ X @ A3 )
@ ^ [X: v] : ( member_v @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_1176_less__eq__set__def,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ord_le5892402249245633078_v_v_o
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ A3 )
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_1177_pred__subset__eq,axiom,
! [R3: set_v,S: set_v] :
( ( ord_less_eq_v_o
@ ^ [X: v] : ( member_v @ X @ R3 )
@ ^ [X: v] : ( member_v @ X @ S ) )
= ( ord_less_eq_set_v @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_1178_pred__subset__eq,axiom,
! [R3: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( ord_le5892402249245633078_v_v_o
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ R3 )
@ ^ [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ S ) )
= ( ord_le7336532860387713383od_v_v @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_1179_bot__empty__eq2,axiom,
( bot_bo2627483263562843604unit_o
= ( ^ [X: v,Y: sCC_Bl1394983891496994913t_unit] : ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X @ Y ) @ bot_bo1868334660695016813t_unit ) ) ) ).
% bot_empty_eq2
thf(fact_1180_bot__empty__eq2,axiom,
( bot_bot_v_v_o
= ( ^ [X: v,Y: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ bot_bo723834152578015283od_v_v ) ) ) ).
% bot_empty_eq2
thf(fact_1181_graph_Oedge__ra,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y4: product_prod_v_v,X3: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y4 @ ( Successors @ X3 ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Y4 ) @ E4 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X3 @ Y4 @ E4 ) ) ) ) ).
% graph.edge_ra
thf(fact_1182_graph_Oedge__ra,axiom,
! [Vertices: set_v,Successors: v > set_v,Y4: v,X3: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y4 @ ( Successors @ X3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y4 ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y4 @ E4 ) ) ) ) ).
% graph.edge_ra
thf(fact_1183_graph_Ora__cases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X3: product_prod_v_v,Y4: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X3 @ Y4 @ E4 )
=> ( ( X3 = Y4 )
| ? [Z4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Z4 @ ( Successors @ X3 ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Z4 ) @ E4 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ Z4 @ Y4 @ E4 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_1184_graph_Ora__cases,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y4 @ E4 )
=> ( ( X3 = Y4 )
| ? [Z4: v] :
( ( member_v @ Z4 @ ( Successors @ X3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Z4 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ Z4 @ Y4 @ E4 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_1185_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,A22: product_prod_v_v,A32: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A22 @ A32 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: product_prod_v_v] :
( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ Y3 @ A32 )
=> ( ( member7453568604450474000od_v_v @ A22 @ ( Successors @ Y3 ) )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ A22 ) @ A32 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_1186_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A22 @ A32 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ Y3 @ A32 )
=> ( ( member_v @ A22 @ ( Successors @ Y3 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ A22 ) @ A32 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_1187_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,A22: product_prod_v_v,A32: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A22 @ A32 )
= ( ? [X: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v] :
( ( A1 = X )
& ( A22 = X )
& ( A32 = E6 ) )
| ? [X: product_prod_v_v,Y: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v,Z3: product_prod_v_v] :
( ( A1 = X )
& ( A22 = Z3 )
& ( A32 = E6 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E6 )
& ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ Z3 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_1188_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A22 @ A32 )
= ( ? [X: v,E6: set_Product_prod_v_v] :
( ( A1 = X )
& ( A22 = X )
& ( A32 = E6 ) )
| ? [X: v,Y: v,E6: set_Product_prod_v_v,Z3: v] :
( ( A1 = X )
& ( A22 = Z3 )
& ( A32 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E6 )
& ( member_v @ Z3 @ ( Successors @ Y ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z3 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_1189_graph_Ora__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X3: product_prod_v_v,Y4: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v,Z2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X3 @ Y4 @ E4 )
=> ( ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y4 ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y4 @ Z2 ) @ E4 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X3 @ Z2 @ E4 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_1190_graph_Ora__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v,E4: set_Product_prod_v_v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y4 @ E4 )
=> ( ( member_v @ Z2 @ ( Successors @ Y4 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y4 @ Z2 ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Z2 @ E4 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_1191_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_v,K: set_v,A: set_v,B3: set_v] :
( ( A2
= ( inf_inf_set_v @ K @ A ) )
=> ( ( inf_inf_set_v @ A2 @ B3 )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1192_boolean__algebra__cancel_Oinf1,axiom,
! [A2: product_unit,K: product_unit,A: product_unit,B3: product_unit] :
( ( A2
= ( inf_inf_Product_unit @ K @ A ) )
=> ( ( inf_inf_Product_unit @ A2 @ B3 )
= ( inf_inf_Product_unit @ K @ ( inf_inf_Product_unit @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1193_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_v,K: set_v,B3: set_v,A: set_v] :
( ( B
= ( inf_inf_set_v @ K @ B3 ) )
=> ( ( inf_inf_set_v @ A @ B )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1194_boolean__algebra__cancel_Oinf2,axiom,
! [B: product_unit,K: product_unit,B3: product_unit,A: product_unit] :
( ( B
= ( inf_inf_Product_unit @ K @ B3 ) )
=> ( ( inf_inf_Product_unit @ A @ B )
= ( inf_inf_Product_unit @ K @ ( inf_inf_Product_unit @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1195_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_v,K: set_v,A: set_v,B3: set_v] :
( ( A2
= ( sup_sup_set_v @ K @ A ) )
=> ( ( sup_sup_set_v @ A2 @ B3 )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1196_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Product_prod_v_v,K: set_Product_prod_v_v,A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( A2
= ( sup_su414716646722978715od_v_v @ K @ A ) )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B3 )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1197_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_set_v,K: set_set_v,A: set_set_v,B3: set_set_v] :
( ( A2
= ( sup_sup_set_set_v @ K @ A ) )
=> ( ( sup_sup_set_set_v @ A2 @ B3 )
= ( sup_sup_set_set_v @ K @ ( sup_sup_set_set_v @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1198_boolean__algebra__cancel_Osup1,axiom,
! [A2: product_unit,K: product_unit,A: product_unit,B3: product_unit] :
( ( A2
= ( sup_sup_Product_unit @ K @ A ) )
=> ( ( sup_sup_Product_unit @ A2 @ B3 )
= ( sup_sup_Product_unit @ K @ ( sup_sup_Product_unit @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1199_boolean__algebra__cancel_Osup2,axiom,
! [B: set_v,K: set_v,B3: set_v,A: set_v] :
( ( B
= ( sup_sup_set_v @ K @ B3 ) )
=> ( ( sup_sup_set_v @ A @ B )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1200_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Product_prod_v_v,K: set_Product_prod_v_v,B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( B
= ( sup_su414716646722978715od_v_v @ K @ B3 ) )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1201_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_v,K: set_set_v,B3: set_set_v,A: set_set_v] :
( ( B
= ( sup_sup_set_set_v @ K @ B3 ) )
=> ( ( sup_sup_set_set_v @ A @ B )
= ( sup_sup_set_set_v @ K @ ( sup_sup_set_set_v @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1202_boolean__algebra__cancel_Osup2,axiom,
! [B: product_unit,K: product_unit,B3: product_unit,A: product_unit] :
( ( B
= ( sup_sup_Product_unit @ K @ B3 ) )
=> ( ( sup_sup_Product_unit @ A @ B )
= ( sup_sup_Product_unit @ K @ ( sup_sup_Product_unit @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1203_graph_Ora__add__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,X3: v,Y4: v,E4: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y4 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y4 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ V @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ W @ Y4 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% graph.ra_add_edge
thf(fact_1204_graph_Oavoiding__explored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,X3: v,Y4: v,E4: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y4 @ E4 )
=> ( ~ ( member_v @ Y4 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y4 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ) ).
% graph.avoiding_explored
thf(fact_1205_boolean__algebra_Odisj__zero__right,axiom,
! [X3: product_unit] :
( ( sup_sup_Product_unit @ X3 @ bot_bot_Product_unit )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1206_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_v] :
( ( sup_sup_set_v @ X3 @ bot_bot_set_v )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1207_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ bot_bo723834152578015283od_v_v )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1208_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ bot_bot_set_set_v )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1209_subset__code_I1_J,axiom,
! [Xs: list_v,B: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B )
= ( ! [X: v] :
( ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( member_v @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1210_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 )
= ( ! [X: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1211_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ Y4 @ Z2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X3 @ Y4 ) @ ( inf_inf_set_v @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1212_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y4 @ Z2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y4 ) @ ( inf_in6271465464967711157od_v_v @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1213_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] :
( ( inf_inf_set_set_v @ X3 @ ( sup_sup_set_set_v @ Y4 @ Z2 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X3 @ Y4 ) @ ( inf_inf_set_set_v @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1214_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1215_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_v,Y4: set_v,Z2: set_v] :
( ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ Y4 @ Z2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X3 @ Y4 ) @ ( sup_sup_set_v @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1216_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y4 @ Z2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y4 ) @ ( sup_su414716646722978715od_v_v @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1217_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_set_v,Y4: set_set_v,Z2: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ ( inf_inf_set_set_v @ Y4 @ Z2 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X3 @ Y4 ) @ ( sup_sup_set_set_v @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1218_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1219_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y4: set_v,Z2: set_v,X3: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y4 @ Z2 ) @ X3 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y4 @ X3 ) @ ( inf_inf_set_v @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1220_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v,X3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y4 @ Z2 ) @ X3 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y4 @ X3 ) @ ( inf_in6271465464967711157od_v_v @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1221_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y4: set_set_v,Z2: set_set_v,X3: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y4 @ Z2 ) @ X3 )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y4 @ X3 ) @ ( inf_inf_set_set_v @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1222_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y4: product_unit,Z2: product_unit,X3: product_unit] :
( ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y4 @ Z2 ) @ X3 )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y4 @ X3 ) @ ( inf_inf_Product_unit @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1223_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y4: set_v,Z2: set_v,X3: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y4 @ Z2 ) @ X3 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y4 @ X3 ) @ ( sup_sup_set_v @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1224_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y4: set_Product_prod_v_v,Z2: set_Product_prod_v_v,X3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y4 @ Z2 ) @ X3 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y4 @ X3 ) @ ( sup_su414716646722978715od_v_v @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1225_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y4: set_set_v,Z2: set_set_v,X3: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y4 @ Z2 ) @ X3 )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y4 @ X3 ) @ ( sup_sup_set_set_v @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1226_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y4: product_unit,Z2: product_unit,X3: product_unit] :
( ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y4 @ Z2 ) @ X3 )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y4 @ X3 ) @ ( sup_sup_Product_unit @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1227_list_Osel_I2_J,axiom,
( ( tl_v @ nil_v )
= nil_v ) ).
% list.sel(2)
thf(fact_1228_diff__shunt__var,axiom,
! [X3: set_set_v,Y4: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ X3 @ Y4 )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ X3 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_1229_diff__shunt__var,axiom,
! [X3: set_v,Y4: set_v] :
( ( ( minus_minus_set_v @ X3 @ Y4 )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ X3 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_1230_diff__shunt__var,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ X3 @ Y4 )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ X3 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_1231_empty__set,axiom,
( bot_bot_set_v
= ( set_v2 @ nil_v ) ) ).
% empty_set
thf(fact_1232_empty__set,axiom,
( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% empty_set
thf(fact_1233_empty__set,axiom,
( bot_bot_set_set_v
= ( set_set_v2 @ nil_set_v ) ) ).
% empty_set
thf(fact_1234_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_1235_hd__in__set,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( member_v @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1236_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_1237_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_1238_list_Oset__sel_I2_J,axiom,
! [A: list_P7986770385144383213od_v_v,X3: product_prod_v_v] :
( ( A != nil_Product_prod_v_v )
=> ( ( member7453568604450474000od_v_v @ X3 @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ A ) ) )
=> ( member7453568604450474000od_v_v @ X3 @ ( set_Product_prod_v_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1239_list_Oset__sel_I2_J,axiom,
! [A: list_v,X3: v] :
( ( A != nil_v )
=> ( ( member_v @ X3 @ ( set_v2 @ ( tl_v @ A ) ) )
=> ( member_v @ X3 @ ( set_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1240_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_1241_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 ) ) ) )
& ! [X: v] :
( ( member_v @ X @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( member_v @ X @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V @ Ns ) ) ) ) ).
% pre_dfss_def
thf(fact_1242_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 ) ) ) )
& ! [X: v] :
( ( member_v @ X @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( member_v @ X @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X: v] :
( ( member_v @ X @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V @ Ns ) ) ) ) ) ).
% graph.pre_dfss_def
thf(fact_1243_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_1244_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_1245_set__union,axiom,
! [Xs: list_set_v,Ys: list_set_v] :
( ( set_set_v2 @ ( union_set_v @ Xs @ Ys ) )
= ( sup_sup_set_set_v @ ( set_set_v2 @ Xs ) @ ( set_set_v2 @ Ys ) ) ) ).
% set_union
thf(fact_1246_equality,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ R )
= ( sCC_Bl1090238580953940555t_unit @ R2 ) )
=> ( ( ( sCC_Bl1280885523602775798t_unit @ R )
= ( sCC_Bl1280885523602775798t_unit @ R2 ) )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R )
= ( sCC_Bl157864678168468314t_unit @ R2 ) )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R )
= ( sCC_Bl4645233313691564917t_unit @ R2 ) )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R )
= ( sCC_Bl3795065053823578884t_unit @ R2 ) )
=> ( ( ( sCC_Bl2536197123907397897t_unit @ R )
= ( sCC_Bl2536197123907397897t_unit @ R2 ) )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ R )
= ( sCC_Bl8828226123343373779t_unit @ R2 ) )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R )
= ( sCC_Bl9201514103433284750t_unit @ R2 ) )
=> ( ( ( sCC_Bl3567736435408124606t_unit @ R )
= ( sCC_Bl3567736435408124606t_unit @ R2 ) )
=> ( R = R2 ) ) ) ) ) ) ) ) ) ) ).
% equality
thf(fact_1247_list_Oinject,axiom,
! [X21: v,X22: list_v,Y21: v,Y22: list_v] :
( ( ( cons_v @ X21 @ X22 )
= ( cons_v @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_1248_list_Osimps_I15_J,axiom,
! [X21: product_prod_v_v,X22: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) )
= ( insert1338601472111419319od_v_v @ X21 @ ( set_Product_prod_v_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1249_list_Osimps_I15_J,axiom,
! [X21: set_v,X22: list_set_v] :
( ( set_set_v2 @ ( cons_set_v @ X21 @ X22 ) )
= ( insert_set_v @ X21 @ ( set_set_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1250_list_Osimps_I15_J,axiom,
! [X21: v,X22: list_v] :
( ( set_v2 @ ( cons_v @ X21 @ X22 ) )
= ( insert_v @ X21 @ ( set_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1251_append1__eq__conv,axiom,
! [Xs: list_v,X3: v,Ys: list_v,Y4: v] :
( ( ( append_v @ Xs @ ( cons_v @ X3 @ nil_v ) )
= ( append_v @ Ys @ ( cons_v @ Y4 @ nil_v ) ) )
= ( ( Xs = Ys )
& ( X3 = Y4 ) ) ) ).
% append1_eq_conv
thf(fact_1252_hd__Cons__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( cons_v @ ( hd_v @ Xs ) @ ( tl_v @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_1253_list_Ocollapse,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) )
= List ) ) ).
% list.collapse
thf(fact_1254_list__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X2: v] : ( P @ ( cons_v @ X2 @ nil_v ) )
=> ( ! [X2: v,Xs3: list_v] :
( ( Xs3 != nil_v )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_v @ X2 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1255_list__induct2_H,axiom,
! [P: list_v > list_v > $o,Xs: list_v,Ys: list_v] :
( ( P @ nil_v @ nil_v )
=> ( ! [X2: v,Xs3: list_v] : ( P @ ( cons_v @ X2 @ Xs3 ) @ nil_v )
=> ( ! [Y3: v,Ys3: list_v] : ( P @ nil_v @ ( cons_v @ Y3 @ Ys3 ) )
=> ( ! [X2: v,Xs3: list_v,Y3: v,Ys3: list_v] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_v @ X2 @ Xs3 ) @ ( cons_v @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1256_neq__Nil__conv,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
= ( ? [Y: v,Ys4: list_v] :
( Xs
= ( cons_v @ Y @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1257_successively_Ocases,axiom,
! [X3: produc8237170675765753490list_v] :
( ! [P6: v > v > $o] :
( X3
!= ( produc601102195597853570list_v @ P6 @ nil_v ) )
=> ( ! [P6: v > v > $o,X2: v] :
( X3
!= ( produc601102195597853570list_v @ P6 @ ( cons_v @ X2 @ nil_v ) ) )
=> ~ ! [P6: v > v > $o,X2: v,Y3: v,Xs3: list_v] :
( X3
!= ( produc601102195597853570list_v @ P6 @ ( cons_v @ X2 @ ( cons_v @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1258_remdups__adj_Ocases,axiom,
! [X3: list_v] :
( ( X3 != nil_v )
=> ( ! [X2: v] :
( X3
!= ( cons_v @ X2 @ nil_v ) )
=> ~ ! [X2: v,Y3: v,Xs3: list_v] :
( X3
!= ( cons_v @ X2 @ ( cons_v @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1259_sorted__wrt_Ocases,axiom,
! [X3: produc8237170675765753490list_v] :
( ! [P6: v > v > $o] :
( X3
!= ( produc601102195597853570list_v @ P6 @ nil_v ) )
=> ~ ! [P6: v > v > $o,X2: v,Ys3: list_v] :
( X3
!= ( produc601102195597853570list_v @ P6 @ ( cons_v @ X2 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1260_transpose_Ocases,axiom,
! [X3: list_list_v] :
( ( X3 != nil_list_v )
=> ( ! [Xss: list_list_v] :
( X3
!= ( cons_list_v @ nil_v @ Xss ) )
=> ~ ! [X2: v,Xs3: list_v,Xss: list_list_v] :
( X3
!= ( cons_list_v @ ( cons_v @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_1261_shuffles_Ocases,axiom,
! [X3: produc1391462591744249447list_v] :
( ! [Ys3: list_v] :
( X3
!= ( produc6795410681906604247list_v @ nil_v @ Ys3 ) )
=> ( ! [Xs3: list_v] :
( X3
!= ( produc6795410681906604247list_v @ Xs3 @ nil_v ) )
=> ~ ! [X2: v,Xs3: list_v,Y3: v,Ys3: list_v] :
( X3
!= ( produc6795410681906604247list_v @ ( cons_v @ X2 @ Xs3 ) @ ( cons_v @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1262_splice_Ocases,axiom,
! [X3: produc1391462591744249447list_v] :
( ! [Ys3: list_v] :
( X3
!= ( produc6795410681906604247list_v @ nil_v @ Ys3 ) )
=> ~ ! [X2: v,Xs3: list_v,Ys3: list_v] :
( X3
!= ( produc6795410681906604247list_v @ ( cons_v @ X2 @ Xs3 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_1263_list_Oexhaust,axiom,
! [Y4: list_v] :
( ( Y4 != nil_v )
=> ~ ! [X212: v,X222: list_v] :
( Y4
!= ( cons_v @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_1264_list_OdiscI,axiom,
! [List: list_v,X21: v,X22: list_v] :
( ( List
= ( cons_v @ X21 @ X22 ) )
=> ( List != nil_v ) ) ).
% list.discI
thf(fact_1265_list_Odistinct_I1_J,axiom,
! [X21: v,X22: list_v] :
( nil_v
!= ( cons_v @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_1266_dfss_Opsimps,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ V @ E ) ) )
=> ( ( sCC_Bloemen_dfss_v @ successors @ V @ E )
= ( if_SCC4926449794303880475t_unit
@ ( ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
= bot_bot_set_v )
@ E
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X: v] :
( if_set_v @ ( X = V )
@ ( sup_sup_set_v
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y: v] : ( member_v @ Y @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y: v] : ( member_v @ Y @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y: v] : ( member_v @ Y @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V
@ ( fChoice_v
@ ^ [Y: v] : ( member_v @ Y @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E ) ) )
@ V )
@ ( insert_v
@ ( fChoice_v
@ ^ [Y: v] : ( member_v @ Y @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ bot_bot_set_v ) )
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y: v] : ( member_v @ Y @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y: v] : ( member_v @ Y @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y: v] : ( member_v @ Y @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V
@ ( fChoice_v
@ ^ [Y: v] : ( member_v @ Y @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E ) ) )
@ X ) )
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [X: v] : ( member_v @ X @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [X: v] : ( member_v @ X @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [X: v] : ( member_v @ X @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V
@ ( fChoice_v
@ ^ [X: v] : ( member_v @ X @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E ) ) ) ) ) ) ) ) ).
% dfss.psimps
thf(fact_1267_dfss_Ocases,axiom,
! [X3: produc5741669702376414499t_unit] :
~ ! [V3: v,E7: sCC_Bl1394983891496994913t_unit] :
( X3
!= ( produc3862955338007567901t_unit @ V3 @ E7 ) ) ).
% dfss.cases
thf(fact_1268_dom,axiom,
accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ va @ ea ) ) ).
% dom
thf(fact_1269_pre__dfs__implies__post__dfs,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V @ E )
=> ( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V @ E ) ) )
=> ( ( sCC_Bl6082031138996704384t_unit @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) )
=> ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ ( sCC_Bloemen_dfs_v @ successors @ V @ E ) ) ) ) ) ).
% pre_dfs_implies_post_dfs
thf(fact_1270_dfs__dfss_Odomintros_I1_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors )
@ ( sum_In5289330923152326972t_unit
@ ( produc3862955338007567901t_unit @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( insert_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ E ) ) ) ) ) )
=> ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V @ E ) ) ) ) ).
% dfs_dfss.domintros(1)
thf(fact_1271_dfs_Opsimps,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V @ E ) ) )
=> ( ( sCC_Bloemen_dfs_v @ successors @ V @ E )
= ( if_SCC4926449794303880475t_unit
@ ( V
= ( hd_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] :
( sup_sup_set_v
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) )
@ V ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] :
( sup_sup_set_set_v
@ ( sCC_Bl2536197123907397897t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( insert_set_v
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) )
@ V )
@ bot_bot_set_set_v ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) ) ).
% dfs.psimps
thf(fact_1272_unite__S__equal,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: 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 ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N3 ) )
& ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) )
& ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ).
% unite_S_equal
thf(fact_1273_unite__def,axiom,
( sCC_Bloemen_unite_v
= ( ^ [V4: v,W2: v,E8: sCC_Bl1394983891496994913t_unit] :
( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( dropWhile_v
@ ^ [X: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ X ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) )
@ ( sCC_Bl3155122997657187039t_unit
@ ^ [Uu: v > set_v,X: v] :
( if_set_v
@ ( member_v @ X
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E8 @ Y ) )
& ( member_v @ Y
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z3: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z3 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z3: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z3 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E8 @ Y ) )
& ( member_v @ Y
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z3: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z3 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z3: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z3 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ E8 @ X ) )
@ E8 ) ) ) ) ).
% unite_def
thf(fact_1274_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_1275_inf__unit__def,axiom,
( inf_inf_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_1276_sup__unit__def,axiom,
( sup_sup_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
% Helper facts (12)
thf(help_fChoice_1_1_fChoice_001tf__v_T,axiom,
! [P: v > $o] :
( ( P @ ( fChoice_v @ P ) )
= ( ? [X5: v] : ( P @ X5 ) ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X3: set_v,Y4: set_v] :
( ( if_set_v @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X3: set_v,Y4: set_v] :
( ( if_set_v @ $true @ X3 @ Y4 )
= X3 ) ).
thf(help_fChoice_1_1_fChoice_001t__Set__Oset_Itf__v_J_T,axiom,
! [P: set_v > $o] :
( ( P @ ( fChoice_set_v @ P ) )
= ( ? [X5: set_v] : ( P @ X5 ) ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Set__Oset_Itf__v_J_J_T,axiom,
! [X3: set_set_v,Y4: set_set_v] :
( ( if_set_set_v @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Set__Oset_Itf__v_J_J_T,axiom,
! [X3: set_set_v,Y4: set_set_v] :
( ( if_set_set_v @ $true @ X3 @ Y4 )
= X3 ) ).
thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_T,axiom,
! [P: product_prod_v_v > $o] :
( ( P @ ( fChoic927883735564035387od_v_v @ P ) )
= ( ? [X5: product_prod_v_v] : ( P @ X5 ) ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( if_set4279007504652509325od_v_v @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X3: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( if_set4279007504652509325od_v_v @ $true @ X3 @ Y4 )
= X3 ) ).
thf(help_If_3_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [X3: sCC_Bl1394983891496994913t_unit,Y4: sCC_Bl1394983891496994913t_unit] :
( ( if_SCC4926449794303880475t_unit @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [X3: sCC_Bl1394983891496994913t_unit,Y4: sCC_Bl1394983891496994913t_unit] :
( ( if_SCC4926449794303880475t_unit @ $true @ X3 @ Y4 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( member_v
@ ( fChoice_v
@ ^ [X: v] : ( member_v @ X @ ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) ) )
@ ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) ) ).
%------------------------------------------------------------------------------