TPTP Problem File: ITP115^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP115^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Lower_Semicontinuous problem prob_96__6247558_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Lower_Semicontinuous/prob_96__6247558_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 412 ( 140 unt; 54 typ; 0 def)
% Number of atoms : 913 ( 256 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3781 ( 78 ~; 5 |; 72 &;3254 @)
% ( 0 <=>; 372 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 97 ( 97 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 51 usr; 3 con; 0-4 aty)
% Number of variables : 1046 ( 63 ^; 917 !; 20 ?;1046 :)
% ( 46 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:25:13.913
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (49)
thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
topolo2117631714pology:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple1035589618norder:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot0__space,type,
topological_t0_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot1__space,type,
topological_t1_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot2__space,type,
topological_t2_space:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo890362671_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo47006728_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Odiscrete__topology,type,
topolo2133971006pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo503727757_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
topolo2135403230pology:
!>[A: $tType] : $o ).
thf(sy_c_Elementary__Topology_Oclosure,type,
elementary_closure:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__quczrylfpw_Olsc__at,type,
lower_582600101lsc_at:
!>[A: $tType,B: $tType] : ( A > ( A > B ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
topolo1751647064n_open:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
topolo507301023within:
!>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
topolo406746546ounded:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_A____,type,
a2: b ).
thf(sy_v_f,type,
f: a > b ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_x0,type,
x0: a ).
% Relevant facts (256)
thf(fact_0__092_060open_062A_A_092_060noteq_062_Af_Ax0_092_060close_062,axiom,
( a2
!= ( f @ x0 ) ) ).
% \<open>A \<noteq> f x0\<close>
thf(fact_1__092_060open_062f_Ax0_A_092_060noteq_062_AA_A_092_060Longrightarrow_062_A_092_060exists_062U_AV_O_Aopen_AU_A_092_060and_062_Aopen_AV_A_092_060and_062_Af_Ax0_A_092_060in_062_AU_A_092_060and_062_AA_A_092_060in_062_AV_A_092_060and_062_AU_A_092_060inter_062_AV_A_061_A_123_125_092_060close_062,axiom,
( ( ( f @ x0 )
!= a2 )
=> ? [U: set @ b,V: set @ b] :
( ( topolo1751647064n_open @ b @ U )
& ( topolo1751647064n_open @ b @ V )
& ( member @ b @ ( f @ x0 ) @ U )
& ( member @ b @ a2 @ V )
& ( ( inf_inf @ ( set @ b ) @ U @ V )
= ( bot_bot @ ( set @ b ) ) ) ) ) ).
% \<open>f x0 \<noteq> A \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> f x0 \<in> U \<and> A \<in> V \<and> U \<inter> V = {}\<close>
thf(fact_2_open__Int,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S: set @ A,T: set @ A] :
( ( topolo1751647064n_open @ A @ S )
=> ( ( topolo1751647064n_open @ A @ T )
=> ( topolo1751647064n_open @ A @ ( inf_inf @ ( set @ A ) @ S @ T ) ) ) ) ) ).
% open_Int
thf(fact_3_open__empty,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ( topolo1751647064n_open @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% open_empty
thf(fact_4_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_5_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_6_hausdorff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ? [U: set @ A,V: set @ A] :
( ( topolo1751647064n_open @ A @ U )
& ( topolo1751647064n_open @ A @ V )
& ( member @ A @ X @ U )
& ( member @ A @ Y @ V )
& ( ( inf_inf @ ( set @ A ) @ U @ V )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% hausdorff
thf(fact_7_separation__t2,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ? [U2: set @ A,V2: set @ A] :
( ( topolo1751647064n_open @ A @ U2 )
& ( topolo1751647064n_open @ A @ V2 )
& ( member @ A @ X @ U2 )
& ( member @ A @ Y @ V2 )
& ( ( inf_inf @ ( set @ A ) @ U2 @ V2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% separation_t2
thf(fact_8_IntI,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ( member @ A @ C @ B2 )
=> ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_9_Int__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C @ A2 )
& ( member @ A @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_10_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X2: A] : ( inf_inf @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% inf_apply
thf(fact_11_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ A3 )
= A3 ) ) ).
% inf.idem
thf(fact_12_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ X )
= X ) ) ).
% inf_idem
thf(fact_13_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B3 ) )
= ( inf_inf @ A @ A3 @ B3 ) ) ) ).
% inf.left_idem
thf(fact_14_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_15_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_16_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_17_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_18_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_right_idem
thf(fact_19_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B3 ) @ B3 )
= ( inf_inf @ A @ A3 @ B3 ) ) ) ).
% inf.right_idem
thf(fact_20_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_left_idem
thf(fact_21__092_060open_062_092_060forall_062S_O_Aopen_AS_A_092_060and_062_Af_Ax0_A_092_060in_062_AS_A_092_060longrightarrow_062_A_I_092_060exists_062T_O_Aopen_AT_A_092_060and_062_Ax0_A_092_060in_062_AT_A_092_060and_062_A_I_092_060forall_062x_H_092_060in_062T_O_Af_Ax_H_A_092_060le_062_Af_Ax0_A_092_060longrightarrow_062_Af_Ax_H_A_092_060in_062_AS_J_J_092_060close_062,axiom,
! [S2: set @ b] :
( ( ( topolo1751647064n_open @ b @ S2 )
& ( member @ b @ ( f @ x0 ) @ S2 ) )
=> ? [T2: set @ a] :
( ( topolo1751647064n_open @ a @ T2 )
& ( member @ a @ x0 @ T2 )
& ! [X3: a] :
( ( member @ a @ X3 @ T2 )
=> ( ( ord_less_eq @ b @ ( f @ X3 ) @ ( f @ x0 ) )
=> ( member @ b @ ( f @ X3 ) @ S2 ) ) ) ) ) ).
% \<open>\<forall>S. open S \<and> f x0 \<in> S \<longrightarrow> (\<exists>T. open T \<and> x0 \<in> T \<and> (\<forall>x'\<in>T. f x' \<le> f x0 \<longrightarrow> f x' \<in> S))\<close>
thf(fact_22_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_23_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_24_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_25_equals0D,axiom,
! [A: $tType,A2: set @ A,A3: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A2 ) ) ).
% equals0D
thf(fact_26_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_27_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% inf_left_commute
thf(fact_28_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A3: A,C: A] :
( ( inf_inf @ A @ B3 @ ( inf_inf @ A @ A3 @ C ) )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% inf.left_commute
thf(fact_29_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% inf_commute
thf(fact_30_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A4: A,B4: A] : ( inf_inf @ A @ B4 @ A4 ) ) ) ) ).
% inf.commute
thf(fact_31_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% inf_assoc
thf(fact_32_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ).
% inf.assoc
thf(fact_33_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( inf_inf @ A @ K @ B3 ) )
=> ( ( inf_inf @ A @ A3 @ B2 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_34_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,K: A,A3: A,B3: A] :
( ( A2
= ( inf_inf @ A @ K @ A3 ) )
=> ( ( inf_inf @ A @ A2 @ B3 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_35_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X2: A] : ( inf_inf @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% inf_fun_def
thf(fact_36_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y3: A] : ( inf_inf @ A @ Y3 @ X2 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_37_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_38_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_39_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_sup_aci(4)
thf(fact_40_Int__left__commute,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
= ( inf_inf @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_41_Int__left__absorb,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_42_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_43_Int__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_44_Int__assoc,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
( ! [X4: A] :
( ( F2 @ X4 )
= ( G2 @ X4 ) )
=> ( F2 = G2 ) ) ).
% ext
thf(fact_49_IntD2,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
=> ( member @ A @ C @ B2 ) ) ).
% IntD2
thf(fact_50_IntD1,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
=> ( member @ A @ C @ A2 ) ) ).
% IntD1
thf(fact_51_IntE,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( ( member @ A @ C @ A2 )
=> ~ ( member @ A @ C @ B2 ) ) ) ).
% IntE
thf(fact_52_open__discrete,axiom,
! [A: $tType] :
( ( topolo2133971006pology @ A )
=> ! [A2: set @ A] : ( topolo1751647064n_open @ A @ A2 ) ) ).
% open_discrete
thf(fact_53_separation__t1,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ? [U2: set @ A] :
( ( topolo1751647064n_open @ A @ U2 )
& ( member @ A @ X @ U2 )
& ~ ( member @ A @ Y @ U2 ) ) ) ) ) ).
% separation_t1
thf(fact_54_separation__t0,axiom,
! [A: $tType] :
( ( topological_t0_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ? [U2: set @ A] :
( ( topolo1751647064n_open @ A @ U2 )
& ( ( member @ A @ X @ U2 )
!= ( member @ A @ Y @ U2 ) ) ) ) ) ) ).
% separation_t0
thf(fact_55_t1__space,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ? [U: set @ A] :
( ( topolo1751647064n_open @ A @ U )
& ( member @ A @ X @ U )
& ~ ( member @ A @ Y @ U ) ) ) ) ).
% t1_space
thf(fact_56_t0__space,axiom,
! [A: $tType] :
( ( topological_t0_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ? [U: set @ A] :
( ( topolo1751647064n_open @ A @ U )
& ( ( member @ A @ X @ U )
!= ( member @ A @ Y @ U ) ) ) ) ) ).
% t0_space
thf(fact_57_disjoint__iff__not__equal,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ B2 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_58_Int__empty__right,axiom,
! [A: $tType,A2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_59_Int__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_60_disjoint__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ~ ( member @ A @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_61_Int__emptyI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ~ ( member @ A @ X4 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_62_bot__apply,axiom,
! [C3: $tType,D: $tType] :
( ( bot @ C3 )
=> ( ( bot_bot @ ( D > C3 ) )
= ( ^ [X2: D] : ( bot_bot @ C3 ) ) ) ) ).
% bot_apply
thf(fact_63_calculation,axiom,
( ~ ! [S2: set @ b] :
( ( ( topolo1751647064n_open @ b @ S2 )
& ( member @ b @ ( f @ x0 ) @ S2 ) )
=> ? [T2: set @ a] :
( ( topolo1751647064n_open @ a @ T2 )
& ( member @ a @ x0 @ T2 )
& ! [X3: a] :
( ( member @ a @ X3 @ T2 )
=> ( ( ord_less_eq @ b @ ( f @ X3 ) @ ( f @ x0 ) )
=> ( member @ b @ ( f @ X3 ) @ S2 ) ) ) ) )
=> ~ ( lower_582600101lsc_at @ a @ b @ x0 @ f ) ) ).
% calculation
thf(fact_64_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: set @ A] :
( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_65_totally__bounded__empty,axiom,
! [A: $tType] :
( ( topolo47006728_space @ A )
=> ( topolo406746546ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% totally_bounded_empty
thf(fact_66_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_67_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_68_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_69_is__singletonI_H,axiom,
! [A: $tType,A2: set @ A] :
( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y2: A] :
( ( member @ A @ X4 @ A2 )
=> ( ( member @ A @ Y2 @ A2 )
=> ( X4 = Y2 ) ) )
=> ( is_singleton @ A @ A2 ) ) ) ).
% is_singletonI'
thf(fact_70_Diff__disjoint,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_71_insert__disjoint_I1_J,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_72_insert__disjoint_I2_J,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ B2 ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_73_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_74_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A5: A > B,B5: A > B,X2: A] : ( minus_minus @ B @ ( A5 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_75_insertCI,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A3 @ B2 )
=> ( A3 = B3 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_76_insert__iff,axiom,
! [A: $tType,A3: A,B3: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
= ( ( A3 = B3 )
| ( member @ A @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_77_insert__absorb2,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
= ( insert @ A @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_78_DiffI,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ~ ( member @ A @ C @ B2 )
=> ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_79_Diff__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C @ A2 )
& ~ ( member @ A @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_80_Diff__idemp,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_81_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% le_inf_iff
thf(fact_82_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) )
= ( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% inf.bounded_iff
thf(fact_83_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_84_Int__insert__left__if0,axiom,
! [A: $tType,A3: A,C2: set @ A,B2: set @ A] :
( ~ ( member @ A @ A3 @ C2 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_85_Int__insert__left__if1,axiom,
! [A: $tType,A3: A,C2: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ C2 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C2 )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_86_insert__inter__insert,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ ( insert @ A @ A3 @ B2 ) )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_87_Int__insert__right__if0,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_88_Int__insert__right__if1,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_89_Diff__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= A2 ) ).
% Diff_empty
thf(fact_90_empty__Diff,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_91_Diff__cancel,axiom,
! [A: $tType,A2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_92_Diff__insert0,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_93_insert__Diff1,axiom,
! [A: $tType,X: A,B2: set @ A,A2: set @ A] :
( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_94_disjoint__insert_I2_J,axiom,
! [A: $tType,A2: set @ A,B3: A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) )
= ( ~ ( member @ A @ B3 @ A2 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_95_disjoint__insert_I1_J,axiom,
! [A: $tType,B2: set @ A,A3: A,A2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ A2 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ B2 @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_96_insert__Diff__single,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A3 @ A2 ) ) ).
% insert_Diff_single
thf(fact_97_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_98_DiffE,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B2 ) ) ) ).
% DiffE
thf(fact_99_DiffD1,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ( member @ A @ C @ A2 ) ) ).
% DiffD1
thf(fact_100_DiffD2,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( member @ A @ C @ B2 ) ) ).
% DiffD2
thf(fact_101_insertE,axiom,
! [A: $tType,A3: A,B3: A,A2: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
=> ( ( A3 != B3 )
=> ( member @ A @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_102_insertI1,axiom,
! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% insertI1
thf(fact_103_insertI2,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( member @ A @ A3 @ B2 )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_104_Set_Oset__insert,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ( member @ A @ X @ A2 )
=> ~ ! [B6: set @ A] :
( ( A2
= ( insert @ A @ X @ B6 ) )
=> ( member @ A @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_105_insert__ident,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ~ ( member @ A @ X @ B2 )
=> ( ( ( insert @ A @ X @ A2 )
= ( insert @ A @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_106_insert__absorb,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_107_insert__eq__iff,axiom,
! [A: $tType,A3: A,A2: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A2 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B3 @ B2 ) )
= ( ( ( A3 = B3 )
=> ( A2 = B2 ) )
& ( ( A3 != B3 )
=> ? [C4: set @ A] :
( ( A2
= ( insert @ A @ B3 @ C4 ) )
& ~ ( member @ A @ B3 @ C4 )
& ( B2
= ( insert @ A @ A3 @ C4 ) )
& ~ ( member @ A @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_108_insert__Diff__if,axiom,
! [A: $tType,X: A,B2: set @ A,A2: set @ A] :
( ( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_109_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A2: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A2 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_110_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A5: A > B,B5: A > B,X2: A] : ( minus_minus @ B @ ( A5 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_111_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ? [B6: set @ A] :
( ( A2
= ( insert @ A @ A3 @ B6 ) )
& ~ ( member @ A @ A3 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_112_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_113_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_114_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_115_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A6: A,B7: A] :
( ( ord_less_eq @ A @ A6 @ B7 )
=> ( P @ A6 @ B7 ) )
=> ( ! [A6: A,B7: A] :
( ( P @ B7 @ A6 )
=> ( P @ A6 @ B7 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_116_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_117_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_118_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_119_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_120_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_121_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_122_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_123_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_124_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% order.trans
thf(fact_125_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_126_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_127_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_128_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_129_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_130_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F2: A > B,C: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_131_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C: B] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X4: B,Y2: B] :
( ( ord_less_eq @ B @ X4 @ Y2 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_132_order__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 )
& ( order @ A ) )
=> ! [A3: A,B3: A,F2: A > C3,C: C3] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C3 @ ( F2 @ B3 ) @ C )
=> ( ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ C3 @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ C3 @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_133_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X4: B,Y2: B] :
( ( ord_less_eq @ B @ X4 @ Y2 )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_134_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F: A > B,G: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_135_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% le_funI
thf(fact_136_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funE
thf(fact_137_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funD
thf(fact_138_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A2: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_139_Diff__insert2,axiom,
! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_140_insert__Diff,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_141_Diff__insert,axiom,
! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_142_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_143_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_144_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_145_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
? [X2: A] :
( A5
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_146_is__singletonE,axiom,
! [A: $tType,A2: set @ A] :
( ( is_singleton @ A @ A2 )
=> ~ ! [X4: A] :
( A2
!= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_147_singletonD,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_148_singleton__iff,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_149_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B3: A,C: A,D2: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C )
& ( B3 = D2 ) )
| ( ( A3 = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_150_insert__not__empty,axiom,
! [A: $tType,A3: A,A2: set @ A] :
( ( insert @ A @ A3 @ A2 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_151_singleton__inject,axiom,
! [A: $tType,A3: A,B3: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_152_Int__insert__left,axiom,
! [A: $tType,A3: A,C2: set @ A,B2: set @ A] :
( ( ( member @ A @ A3 @ C2 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C2 )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) )
& ( ~ ( member @ A @ A3 @ C2 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_153_Int__insert__right,axiom,
! [A: $tType,A3: A,A2: set @ A,B2: set @ A] :
( ( ( member @ A @ A3 @ A2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) )
& ( ~ ( member @ A @ A3 @ A2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_154_Int__Diff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_155_Diff__Int2,axiom,
! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_156_Diff__Diff__Int,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_157_Diff__Int__distrib,axiom,
! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C2 @ A2 ) @ ( inf_inf @ ( set @ A ) @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_158_Diff__Int__distrib2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ C2 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ C2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_159_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_160_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_sup_ord(1)
thf(fact_161_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_le1
thf(fact_162_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_163_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A3 )
=> ~ ( ord_less_eq @ A @ X @ B3 ) ) ) ) ).
% le_infE
thf(fact_164_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ X @ B3 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% le_infI
thf(fact_165_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C )
=> ( ( ord_less_eq @ A @ B3 @ D2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ ( inf_inf @ A @ C @ D2 ) ) ) ) ) ).
% inf_mono
thf(fact_166_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% le_infI1
thf(fact_167_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,X: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ X ) ) ) ).
% le_infI2
thf(fact_168_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3
= ( inf_inf @ A @ A3 @ B3 ) ) ) ) ).
% inf.orderE
thf(fact_169_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( inf_inf @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% inf.orderI
thf(fact_170_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F2: A > A > A,X: A,Y: A] :
( ! [X4: A,Y2: A] : ( ord_less_eq @ A @ ( F2 @ X4 @ Y2 ) @ X4 )
=> ( ! [X4: A,Y2: A] : ( ord_less_eq @ A @ ( F2 @ X4 @ Y2 ) @ Y2 )
=> ( ! [X4: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ( ord_less_eq @ A @ X4 @ Z3 )
=> ( ord_less_eq @ A @ X4 @ ( F2 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf @ A @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_171_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y3: A] :
( ( inf_inf @ A @ X2 @ Y3 )
= X2 ) ) ) ) ).
% le_iff_inf
thf(fact_172_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( inf_inf @ A @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb1
thf(fact_173_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb2
thf(fact_174_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( inf_inf @ A @ X @ Y )
= X ) ) ) ).
% inf_absorb1
thf(fact_175_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( inf_inf @ A @ X @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_176_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B3 )
=> ~ ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% inf.boundedE
thf(fact_177_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ A3 @ C )
=> ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B3 @ C ) ) ) ) ) ).
% inf.boundedI
thf(fact_178_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Z )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ) ).
% inf_greatest
thf(fact_179_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( A4
= ( inf_inf @ A @ A4 @ B4 ) ) ) ) ) ).
% inf.order_iff
thf(fact_180_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ A3 ) ) ).
% inf.cobounded1
thf(fact_181_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ B3 ) ) ).
% inf.cobounded2
thf(fact_182_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( ( inf_inf @ A @ A4 @ B4 )
= A4 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_183_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( ( inf_inf @ A @ A4 @ B4 )
= B4 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_184_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ C )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C ) ) ) ).
% inf.coboundedI1
thf(fact_185_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,C: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B3 ) @ C ) ) ) ).
% inf.coboundedI2
thf(fact_186_not__open__singleton,axiom,
! [A: $tType] :
( ( topolo890362671_space @ A )
=> ! [X: A] :
~ ( topolo1751647064n_open @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% not_open_singleton
thf(fact_187_Diff__triv,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_188_open__delete,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [S3: set @ A,X: A] :
( ( topolo1751647064n_open @ A @ S3 )
=> ( topolo1751647064n_open @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% open_delete
thf(fact_189_Int__Diff__disjoint,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_190_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
( A5
= ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_191_ball__insert,axiom,
! [A: $tType,A3: A,B2: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( insert @ A @ A3 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ( P @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_insert
thf(fact_192_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_193_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
= ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_194_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_195_subset__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_196_insert__subset,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
= ( ( member @ A @ X @ B2 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_197_Int__subset__iff,axiom,
! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C2 @ A2 )
& ( ord_less_eq @ ( set @ A ) @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_198_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A3: A,A2: set @ A] :
( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A2 ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_199_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A2: set @ A,B3: A] :
( ( ( insert @ A @ A3 @ A2 )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_200_Diff__eq__empty__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_201_subset__insertI2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_202_subset__insertI,axiom,
! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% subset_insertI
thf(fact_203_subset__insert,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_204_insert__mono,axiom,
! [A: $tType,C2: set @ A,D3: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C2 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% insert_mono
thf(fact_205_Int__Collect__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B2 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_206_Int__greatest,axiom,
! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ C2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_207_Int__absorb2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_208_Int__absorb1,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_209_Int__lower2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_210_Int__lower1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_211_Int__mono,axiom,
! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) @ ( inf_inf @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_212_first__countable__basis,axiom,
! [A: $tType] :
( ( topolo2135403230pology @ A )
=> ! [X: A] :
? [A7: nat > ( set @ A )] :
( ! [I: nat] :
( ( member @ A @ X @ ( A7 @ I ) )
& ( topolo1751647064n_open @ A @ ( A7 @ I ) ) )
& ! [S2: set @ A] :
( ( ( topolo1751647064n_open @ A @ S2 )
& ( member @ A @ X @ S2 ) )
=> ? [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( A7 @ I2 ) @ S2 ) ) ) ) ).
% first_countable_basis
thf(fact_213_open__subopen,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ( ( topolo1751647064n_open @ A )
= ( ^ [S4: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S4 )
=> ? [T3: set @ A] :
( ( topolo1751647064n_open @ A @ T3 )
& ( member @ A @ X2 @ T3 )
& ( ord_less_eq @ ( set @ A ) @ T3 @ S4 ) ) ) ) ) ) ).
% open_subopen
thf(fact_214_topological__space__class_OopenI,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ? [T4: set @ A] :
( ( topolo1751647064n_open @ A @ T4 )
& ( member @ A @ X4 @ T4 )
& ( ord_less_eq @ ( set @ A ) @ T4 @ S ) ) )
=> ( topolo1751647064n_open @ A @ S ) ) ) ).
% topological_space_class.openI
thf(fact_215_double__diff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_216_Diff__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_217_Diff__mono,axiom,
! [A: $tType,A2: set @ A,C2: set @ A,D3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ D3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_218_totally__bounded__subset,axiom,
! [A: $tType] :
( ( topolo47006728_space @ A )
=> ! [S: set @ A,T: set @ A] :
( ( topolo406746546ounded @ A @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T @ S )
=> ( topolo406746546ounded @ A @ T ) ) ) ) ).
% totally_bounded_subset
thf(fact_219_subset__singletonD,axiom,
! [A: $tType,A2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A2
= ( bot_bot @ ( set @ A ) ) )
| ( A2
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_220_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_221_subset__Diff__insert,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ ( insert @ A @ X @ C2 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ C2 ) )
& ~ ( member @ A @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_222_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( A3 = B3 )
= ( C = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_223_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A3: A,C: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_224_Diff__single__insert,axiom,
! [A: $tType,A2: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_225_subset__insert__iff,axiom,
! [A: $tType,A2: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_226_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,D2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ D2 @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_227_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_228_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% diff_right_mono
thf(fact_229_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X2: A,A5: set @ A] : ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_230_at__within__nhd,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [X: A,S: set @ A,T: set @ A,U3: set @ A] :
( ( member @ A @ X @ S )
=> ( ( topolo1751647064n_open @ A @ S )
=> ( ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ T @ S ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U3 @ S ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( topolo507301023within @ A @ X @ T )
= ( topolo507301023within @ A @ X @ U3 ) ) ) ) ) ) ).
% at_within_nhd
thf(fact_231_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A2 )
=> ( member @ A @ X4 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_232_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_233_member__remove,axiom,
! [A: $tType,X: A,Y: A,A2: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y @ A2 ) )
= ( ( member @ A @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_234_at__within__empty,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [A3: A] :
( ( topolo507301023within @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ) ).
% at_within_empty
thf(fact_235_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_236_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B2 ) ) ) ).
% subsetD
thf(fact_237_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_238_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ A @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_239_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_240_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_241_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [T5: A] :
( ( member @ A @ T5 @ A5 )
=> ( member @ A @ T5 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_242_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_243_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_244_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_245_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z2: set @ A] : Y4 = Z2 )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_246_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_247_at__le,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S3: set @ A,T6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ T6 )
=> ( ord_less_eq @ ( filter @ A ) @ ( topolo507301023within @ A @ X @ S3 ) @ ( topolo507301023within @ A @ X @ T6 ) ) ) ) ).
% at_le
thf(fact_248_at__discrete,axiom,
! [A: $tType] :
( ( topolo2133971006pology @ A )
=> ( ( topolo507301023within @ A )
= ( ^ [X2: A,S4: set @ A] : ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% at_discrete
thf(fact_249_insert__subsetI,axiom,
! [A: $tType,X: A,A2: set @ A,X5: set @ A] :
( ( member @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_250_subset__emptyI,axiom,
! [A: $tType,A2: set @ A] :
( ! [X4: A] :
~ ( member @ A @ X4 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_251_at__eq__bot__iff,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [A3: A] :
( ( ( topolo507301023within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) )
= ( topolo1751647064n_open @ A @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_eq_bot_iff
thf(fact_252_at__within__eq__bot__iff,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [C: A,A2: set @ A] :
( ( ( topolo507301023within @ A @ C @ A2 )
= ( bot_bot @ ( filter @ A ) ) )
= ( ~ ( member @ A @ C @ ( elementary_closure @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% at_within_eq_bot_iff
thf(fact_253_top__apply,axiom,
! [C3: $tType,D: $tType] :
( ( top @ C3 )
=> ( ( top_top @ ( D > C3 ) )
= ( ^ [X2: D] : ( top_top @ C3 ) ) ) ) ).
% top_apply
thf(fact_254_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_255_closure__closure,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S: set @ A] :
( ( elementary_closure @ A @ ( elementary_closure @ A @ S ) )
= ( elementary_closure @ A @ S ) ) ) ).
% closure_closure
% Subclasses (23)
thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Obounded__lattice,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ( bounded_lattice @ A ) ) ).
thf(subcl_Complete__Lattices_Ocomplete__linorder___HOL_Otype,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ( type @ A ) ) ).
thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Obot,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ( bot @ A ) ) ).
thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Oord,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ( ord @ A ) ) ).
thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Otop,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ( top @ A ) ) ).
thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Oorder,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ( order @ A ) ) ).
thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Olattice,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ( lattice @ A ) ) ).
thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Olinorder,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ( linorder @ A ) ) ).
thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ( preorder @ A ) ) ).
thf(subcl_Complete__Lattices_Ocomplete__linorder___Orderings_Oorder__bot,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ( order_bot @ A ) ) ).
thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Osemilattice__inf,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ( semilattice_inf @ A ) ) ).
thf(subcl_Complete__Lattices_Ocomplete__linorder___Lattices_Obounded__lattice__bot,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ( bounded_lattice_bot @ A ) ) ).
thf(subcl_Topological__Spaces_Olinorder__topology___HOL_Otype,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ( type @ A ) ) ).
thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Oord,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ( ord @ A ) ) ).
thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Oorder,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ( order @ A ) ) ).
thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Olinorder,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ( linorder @ A ) ) ).
thf(subcl_Topological__Spaces_Olinorder__topology___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ( preorder @ A ) ) ).
thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Ot0__space,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ( topological_t0_space @ A ) ) ).
thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Ot1__space,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ( topological_t1_space @ A ) ) ).
thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Ot2__space,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ( topological_t2_space @ A ) ) ).
thf(subcl_Topological__Spaces_Olinorder__topology___Topological__Spaces_Otopological__space,axiom,
! [A: $tType] :
( ( topolo2117631714pology @ A )
=> ( topolo503727757_space @ A ) ) ).
thf(subcl_Topological__Spaces_Ofirst__countable__topology___HOL_Otype,axiom,
! [A: $tType] :
( ( topolo2135403230pology @ A )
=> ( type @ A ) ) ).
thf(subcl_Topological__Spaces_Ofirst__countable__topology___Topological__Spaces_Otopological__space,axiom,
! [A: $tType] :
( ( topolo2135403230pology @ A )
=> ( topolo503727757_space @ A ) ) ).
% Type constructors (74)
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice,axiom,
! [A8: $tType] : ( bounded_lattice @ ( filter @ A8 ) ) ).
thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
countable @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_1,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Countable_Ocountable_2,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( countable @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_3,axiom,
! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
thf(tcon_Nat_Onat___Countable_Ocountable_4,axiom,
countable @ nat ).
thf(tcon_fun___Countable_Ocountable_5,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( countable @ A9 ) )
=> ( countable @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_6,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounded_lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_7,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( finite_finite @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_8,axiom,
finite_finite @ $o ).
thf(tcon_fun___Topological__Spaces_Ofirst__countable__topology,axiom,
! [A8: $tType,A9: $tType] :
( ( ( countable @ A8 )
& ( topolo2135403230pology @ A9 ) )
=> ( topolo2135403230pology @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
! [A8: $tType,A9: $tType] :
( ( topolo503727757_space @ A9 )
=> ( topolo503727757_space @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounded_lattice_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_inf @ A9 )
=> ( semilattice_inf @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( order_bot @ A9 )
=> ( order_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 )
=> ( lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 )
=> ( top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 )
=> ( bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A8: $tType,A9: $tType] :
( ( minus @ A9 )
=> ( minus @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Topological__Spaces_Ofirst__countable__topology_9,axiom,
topolo2135403230pology @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_10,axiom,
topolo503727757_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology,axiom,
topolo2133971006pology @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space,axiom,
topological_t1_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot0__space,axiom,
topological_t0_space @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_11,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_12,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_13,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_14,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_15,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_16,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_17,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_18,axiom,
minus @ nat ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_19,axiom,
! [A8: $tType] : ( bounded_lattice_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_20,axiom,
! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_21,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_22,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_23,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_24,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_25,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_26,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_27,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_28,axiom,
! [A8: $tType] : ( minus @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_29,axiom,
topolo503727757_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_30,axiom,
topolo2133971006pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_31,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_32,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_33,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot0__space_34,axiom,
topological_t0_space @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_35,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_36,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_37,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_38,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_39,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_40,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_41,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_42,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_43,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_44,axiom,
minus @ $o ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_45,axiom,
! [A8: $tType] : ( bounded_lattice_bot @ ( filter @ A8 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_46,axiom,
! [A8: $tType] : ( semilattice_inf @ ( filter @ A8 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_47,axiom,
! [A8: $tType] : ( order_bot @ ( filter @ A8 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_48,axiom,
! [A8: $tType] : ( preorder @ ( filter @ A8 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_49,axiom,
! [A8: $tType] : ( lattice @ ( filter @ A8 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_50,axiom,
! [A8: $tType] : ( order @ ( filter @ A8 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_51,axiom,
! [A8: $tType] : ( top @ ( filter @ A8 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_52,axiom,
! [A8: $tType] : ( ord @ ( filter @ A8 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_53,axiom,
! [A8: $tType] : ( bot @ ( filter @ A8 ) ) ).
% Free types (3)
thf(tfree_0,hypothesis,
comple1035589618norder @ b ).
thf(tfree_1,hypothesis,
topolo2117631714pology @ b ).
thf(tfree_2,hypothesis,
topolo2135403230pology @ a ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [S2: set @ b,V3: set @ b] :
( ( ( topolo1751647064n_open @ b @ S2 )
& ( topolo1751647064n_open @ b @ V3 )
& ( member @ b @ ( f @ x0 ) @ S2 )
& ( member @ b @ a2 @ V3 )
& ( ( inf_inf @ ( set @ b ) @ S2 @ V3 )
= ( bot_bot @ ( set @ b ) ) ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------