ITP001 Axioms: ITP077_5.ax
%------------------------------------------------------------------------------
% File : ITP077_5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : topology_2.ax [Gau20]
% : HL4077_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 97 ( 10 unt; 16 typ; 0 def)
% Number of atoms : 1651 ( 35 equ)
% Maximal formula atoms : 17 ( 17 avg)
% Number of connectives : 346 ( 3 ~; 0 |; 53 &)
% ( 23 <=>; 267 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 1227 (1227 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 24 ( 16 >; 8 *; 0 +; 0 <<)
% Number of predicates : 25 ( 24 usr; 3 prp; 0-2 aty)
% Number of functors : 16 ( 16 usr; 0 con; 1-3 aty)
% Number of variables : 292 ( 284 !; 8 ?; 292 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_ty_2Etopology_2Etopology,type,
ty_2Etopology_2Etopology: del > del ).
tff(tp_c_2Etopology_2Eclosed,type,
c_2Etopology_2Eclosed: del > $i ).
tff(mem_c_2Etopology_2Eclosed,axiom,
! [A_27a: del] : mem(c_2Etopology_2Eclosed(A_27a),arr(ty_2Etopology_2Etopology(A_27a),bool)) ).
tff(tp_c_2Etopology_2Eclosed__in,type,
c_2Etopology_2Eclosed__in: del > $i ).
tff(mem_c_2Etopology_2Eclosed__in,axiom,
! [A_27a: del] : mem(c_2Etopology_2Eclosed__in(A_27a),arr(ty_2Etopology_2Etopology(A_27a),arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Etopology_2Ehull,type,
c_2Etopology_2Ehull: del > $i ).
tff(mem_c_2Etopology_2Ehull,axiom,
! [A_27a: del] : mem(c_2Etopology_2Ehull(A_27a),arr(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),arr(A_27a,bool)))) ).
tff(tp_c_2Etopology_2Eistopology,type,
c_2Etopology_2Eistopology: del > $i ).
tff(mem_c_2Etopology_2Eistopology,axiom,
! [A_27a: del] : mem(c_2Etopology_2Eistopology(A_27a),arr(arr(arr(A_27a,bool),bool),bool)) ).
tff(tp_c_2Etopology_2Elimpt,type,
c_2Etopology_2Elimpt: del > $i ).
tff(mem_c_2Etopology_2Elimpt,axiom,
! [A_27a: del] : mem(c_2Etopology_2Elimpt(A_27a),arr(ty_2Etopology_2Etopology(A_27a),arr(A_27a,arr(arr(A_27a,bool),bool)))) ).
tff(tp_c_2Etopology_2Eneigh,type,
c_2Etopology_2Eneigh: del > $i ).
tff(mem_c_2Etopology_2Eneigh,axiom,
! [A_27a: del] : mem(c_2Etopology_2Eneigh(A_27a),arr(ty_2Etopology_2Etopology(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),A_27a),bool))) ).
tff(tp_c_2Etopology_2Eopen,type,
c_2Etopology_2Eopen: del > $i ).
tff(mem_c_2Etopology_2Eopen,axiom,
! [A_27a: del] : mem(c_2Etopology_2Eopen(A_27a),arr(ty_2Etopology_2Etopology(A_27a),bool)) ).
tff(tp_c_2Etopology_2Eopen__in,type,
c_2Etopology_2Eopen__in: del > $i ).
tff(mem_c_2Etopology_2Eopen__in,axiom,
! [A_27a: del] : mem(c_2Etopology_2Eopen__in(A_27a),arr(ty_2Etopology_2Etopology(A_27a),arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Etopology_2Etopology,type,
c_2Etopology_2Etopology: del > $i ).
tff(mem_c_2Etopology_2Etopology,axiom,
! [A_27a: del] : mem(c_2Etopology_2Etopology(A_27a),arr(arr(arr(A_27a,bool),bool),ty_2Etopology_2Etopology(A_27a))) ).
tff(tp_c_2Etopology_2Etopspace,type,
c_2Etopology_2Etopspace: del > $i ).
tff(mem_c_2Etopology_2Etopspace,axiom,
! [A_27a: del] : mem(c_2Etopology_2Etopspace(A_27a),arr(ty_2Etopology_2Etopology(A_27a),arr(A_27a,bool))) ).
tff(ax_thm_2Etopology_2Eistopology,axiom,
! [A_27a: del,V0L: $i] :
( mem(V0L,arr(arr(A_27a,bool),bool))
=> ( p(ap(c_2Etopology_2Eistopology(A_27a),V0L))
<=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),V0L))
& ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),V0L))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),V0L)) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),V0L)) ) ) )
& ! [V3k: $i] :
( mem(V3k,arr(arr(A_27a,bool),bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V3k),V0L))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V3k)),V0L)) ) ) ) ) ) ).
tff(ax_thm_2Etopology_2Etopology__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( mem(V0rep,arr(ty_2Etopology_2Etopology(A_27a),arr(arr(A_27a,bool),bool)))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(arr(arr(A_27a,bool),bool),ty_2Etopology_2Etopology(A_27a)),c_2Etopology_2Eistopology(A_27a)),V0rep)) ) ).
tff(ax_thm_2Etopology_2Etopology__tybij,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( mem(V0a,ty_2Etopology_2Etopology(A_27a))
=> ( ap(c_2Etopology_2Etopology(A_27a),ap(c_2Etopology_2Eopen__in(A_27a),V0a)) = V0a ) )
& ! [V1r: $i] :
( mem(V1r,arr(arr(A_27a,bool),bool))
=> ( p(ap(c_2Etopology_2Eistopology(A_27a),V1r))
<=> ( ap(c_2Etopology_2Eopen__in(A_27a),ap(c_2Etopology_2Etopology(A_27a),V1r)) = V1r ) ) ) ) ).
tff(conj_thm_2Etopology_2EISTOPOLOGY__OPEN__IN,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> p(ap(c_2Etopology_2Eistopology(A_27a),ap(c_2Etopology_2Eopen__in(A_27a),V0top))) ) ).
tff(conj_thm_2Etopology_2ETOPOLOGY__EQ,axiom,
! [A_27a: del,V0top1: $i] :
( mem(V0top1,ty_2Etopology_2Etopology(A_27a))
=> ! [V1top2: $i] :
( mem(V1top2,ty_2Etopology_2Etopology(A_27a))
=> ( ( V0top1 = V1top2 )
<=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top1),V2s))
<=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V1top2),V2s)) ) ) ) ) ) ).
tff(ax_thm_2Etopology_2Eopen__DEF,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(c_2Etopology_2Eopen(A_27a),V0s))
<=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0s),c_2Epred__set_2EUNIV(A_27a))) ) ) ).
tff(lamtp_f2014,type,
f2014: ( del * $i ) > $i ).
tff(lameq_f2014,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] : ( ap(f2014(A_27a,V0top),V1s) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),bool),V1s),ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V1s)) ) ) ).
tff(ax_thm_2Etopology_2Etopspace,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ( ap(c_2Etopology_2Etopspace(A_27a),V0top) = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f2014(A_27a,V0top))) ) ) ).
tff(conj_thm_2Etopology_2Eopen__topspace,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(c_2Etopology_2Eopen(A_27a),V0top))
=> ( ap(c_2Etopology_2Etopspace(A_27a),V0top) = c_2Epred__set_2EUNIV(A_27a) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__IN__CLAUSES,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),c_2Epred__set_2EEMPTY(A_27a)))
& ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V1s))
& p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V2t)) )
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t))) ) ) )
& ! [V3k: $i] :
( mem(V3k,arr(arr(A_27a,bool),bool))
=> ( ! [V4s: $i] :
( mem(V4s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4s),V3k))
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V4s)) ) )
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),ap(c_2Epred__set_2EBIGUNION(A_27a),V3k))) ) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__IN__SUBSET,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V1s))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Etopology_2Etopspace(A_27a),V0top))) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__IN__EMPTY,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),c_2Epred__set_2EEMPTY(A_27a))) ) ).
tff(conj_thm_2Etopology_2EOPEN__IN__INTER,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V1s))
& p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V2t)) )
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t))) ) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__IN__BIGUNION,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1k: $i] :
( mem(V1k,arr(arr(A_27a,bool),bool))
=> ( ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),V1k))
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V2s)) ) )
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),ap(c_2Epred__set_2EBIGUNION(A_27a),V1k))) ) ) ) ).
tff(conj_thm_2Etopology_2EBIGUNION__2,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27a,bool))
=> ( ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V0s),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V1t),c_2Epred__set_2EEMPTY(arr(A_27a,bool))))) = ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__IN__UNION,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V1s))
& p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V2t)) )
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))) ) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__IN__TOPSPACE,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),ap(c_2Etopology_2Etopspace(A_27a),V0top))) ) ).
tff(conj_thm_2Etopology_2EOPEN__IN__BIGINTER,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1s))
& ( V1s != c_2Epred__set_2EEMPTY(arr(A_27a,bool)) )
& ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),V1s))
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V2t)) ) ) )
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),ap(c_2Epred__set_2EBIGINTER(A_27a),V1s))) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__IN__SUBOPEN,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V1s))
<=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ? [V3t: $i] :
( mem(V3t,arr(A_27a,bool))
& p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V3t))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V3t))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V3t),V1s)) ) ) ) ) ) ) ).
tff(ax_thm_2Etopology_2Eneigh,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1N: $i] :
( mem(V1N,arr(A_27a,bool))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Etopology_2Eneigh(A_27a),V0top),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),A_27a),V1N),V2x)))
<=> ? [V3P: $i] :
( mem(V3P,arr(A_27a,bool))
& p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V3P))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V3P),V1N))
& p(ap(V3P,V2x)) ) ) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__OWN__NEIGH,axiom,
! [A_27a: del,V0S_27: $i] :
( mem(V0S_27,arr(A_27a,bool))
=> ! [V1top: $i] :
( mem(V1top,ty_2Etopology_2Etopology(A_27a))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V1top),V0S_27))
& p(ap(V0S_27,V2x)) )
=> p(ap(ap(c_2Etopology_2Eneigh(A_27a),V1top),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),A_27a),V0S_27),V2x))) ) ) ) ) ).
tff(lamtp_f2015,type,
f2015: ( del * $i * $i ) > $i ).
tff(lameq_f2015,axiom,
! [A_27a: del,V0S_27: $i] :
( mem(V0S_27,arr(A_27a,bool))
=> ! [V1top: $i] :
( mem(V1top,ty_2Etopology_2Etopology(A_27a))
=> ! [V2P: $i] : ( ap(f2015(A_27a,V0S_27,V1top),V2P) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),bool),V2P),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Etopology_2Eopen__in(A_27a),V1top),V2P)),ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2P),V0S_27))) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__UNOPEN,axiom,
! [A_27a: del,V0S_27: $i] :
( mem(V0S_27,arr(A_27a,bool))
=> ! [V1top: $i] :
( mem(V1top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V1top),V0S_27))
<=> ( ap(c_2Epred__set_2EBIGUNION(A_27a),ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f2015(A_27a,V0S_27,V1top))) = V0S_27 ) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__SUBOPEN,axiom,
! [A_27a: del,V0S_27: $i] :
( mem(V0S_27,arr(A_27a,bool))
=> ! [V1top: $i] :
( mem(V1top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V1top),V0S_27))
<=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(V0S_27,V2x))
=> ? [V3P: $i] :
( mem(V3P,arr(A_27a,bool))
& p(ap(V3P,V2x))
& p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V1top),V3P))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V3P),V0S_27)) ) ) ) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__NEIGH,axiom,
! [A_27a: del,V0S_27: $i] :
( mem(V0S_27,arr(A_27a,bool))
=> ! [V1top: $i] :
( mem(V1top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V1top),V0S_27))
<=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(V0S_27,V2x))
=> ? [V3N: $i] :
( mem(V3N,arr(A_27a,bool))
& p(ap(ap(c_2Etopology_2Eneigh(A_27a),V1top),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),A_27a),V3N),V2x)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V3N),V0S_27)) ) ) ) ) ) ) ).
tff(ax_thm_2Etopology_2Eclosed__in,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V1s))
<=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Etopology_2Etopspace(A_27a),V0top)))
& p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Etopology_2Etopspace(A_27a),V0top)),V1s))) ) ) ) ) ).
tff(ax_thm_2Etopology_2Eclosed__DEF,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(c_2Etopology_2Eclosed(A_27a),V0s))
<=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0s),c_2Epred__set_2EUNIV(A_27a))) ) ) ).
tff(conj_thm_2Etopology_2Eclosed__topspace,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(c_2Etopology_2Eclosed(A_27a),V0top))
=> ( ap(c_2Etopology_2Etopspace(A_27a),V0top) = c_2Epred__set_2EUNIV(A_27a) ) ) ) ).
tff(conj_thm_2Etopology_2ECLOSED__IN__OPEN__IN__COMPL,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(c_2Etopology_2Eclosed(A_27a),V0top))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V1s))
<=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),ap(c_2Epred__set_2ECOMPL(A_27a),V1s))) ) ) ) ) ).
tff(conj_thm_2Etopology_2ECLOSED__IN__SUBSET,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V1s))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Etopology_2Etopspace(A_27a),V0top))) ) ) ) ).
tff(conj_thm_2Etopology_2ECLOSED__IN__EMPTY,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),c_2Epred__set_2EEMPTY(A_27a))) ) ).
tff(conj_thm_2Etopology_2ECLOSED__IN__TOPSPACE,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),ap(c_2Etopology_2Etopspace(A_27a),V0top))) ) ).
tff(conj_thm_2Etopology_2ECLOSED__IN__UNION,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V1s))
& p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V2t)) )
=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))) ) ) ) ) ).
tff(conj_thm_2Etopology_2ECLOSED__IN__BIGINTER,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1k: $i] :
( mem(V1k,arr(arr(A_27a,bool),bool))
=> ( ( ( V1k != c_2Epred__set_2EEMPTY(arr(A_27a,bool)) )
& ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),V1k))
=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V2s)) ) ) )
=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),ap(c_2Epred__set_2EBIGINTER(A_27a),V1k))) ) ) ) ).
tff(conj_thm_2Etopology_2EBIGINTER__2,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27a,bool))
=> ( ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V0s),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V1t),c_2Epred__set_2EEMPTY(arr(A_27a,bool))))) = ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t) ) ) ) ).
tff(conj_thm_2Etopology_2ECLOSED__IN__INTER,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V1s))
& p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V2t)) )
=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t))) ) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__IN__CLOSED__IN__EQ,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V1s))
<=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Etopology_2Etopspace(A_27a),V0top)))
& p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Etopology_2Etopspace(A_27a),V0top)),V1s))) ) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__IN__CLOSED__IN,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Etopology_2Etopspace(A_27a),V0top)))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V1s))
<=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Etopology_2Etopspace(A_27a),V0top)),V1s))) ) ) ) ) ).
tff(conj_thm_2Etopology_2EOPEN__IN__DIFF,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V1s))
& p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V2t)) )
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t))) ) ) ) ) ).
tff(conj_thm_2Etopology_2ECLOSED__IN__DIFF,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V1s))
& p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V2t)) )
=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t))) ) ) ) ) ).
tff(conj_thm_2Etopology_2ECLOSED__IN__BIGUNION,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s: $i] :
( mem(V1s,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1s))
& ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),V1s))
=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V2t)) ) ) )
=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),ap(c_2Epred__set_2EBIGUNION(A_27a),V1s))) ) ) ) ).
tff(ax_thm_2Etopology_2Elimpt,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2S_27: $i] :
( mem(V2S_27,arr(A_27a,bool))
=> ( p(ap(ap(ap(c_2Etopology_2Elimpt(A_27a),V0top),V1x),V2S_27))
<=> ! [V3N: $i] :
( mem(V3N,arr(A_27a,bool))
=> ( p(ap(ap(c_2Etopology_2Eneigh(A_27a),V0top),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),A_27a),V3N),V1x)))
=> ? [V4y: $i] :
( mem(V4y,A_27a)
& ( V1x != V4y )
& p(ap(V2S_27,V4y))
& p(ap(V3N,V4y)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Etopology_2ECLOSED__LIMPT,axiom,
! [A_27a: del,V0top: $i] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(c_2Etopology_2Eclosed(A_27a),V0top))
=> ! [V1S_27: $i] :
( mem(V1S_27,arr(A_27a,bool))
=> ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V1S_27))
<=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(ap(c_2Etopology_2Elimpt(A_27a),V0top),V2x),V1S_27))
=> p(ap(V1S_27,V2x)) ) ) ) ) ) ) ).
tff(lamtp_f2016,type,
f2016: ( del * $i * $i ) > $i ).
tff(lameq_f2016,axiom,
! [A_27a: del,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V2t: $i] : ( ap(f2016(A_27a,V1s,V0P),V2t) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),bool),V2t),ap(ap(c_2Ebool_2E_2F_5C,ap(V0P,V2t)),ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t))) ) ) ) ).
tff(ax_thm_2Etopology_2Ehull,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s) = ap(c_2Epred__set_2EBIGINTER(A_27a),ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f2016(A_27a,V1s,V0P))) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__P,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(V0P,V1s))
=> ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s) = V1s ) ) ) ) ).
tff(conj_thm_2Etopology_2EP__HULL,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2f: $i] :
( mem(V2f,arr(arr(A_27a,bool),bool))
=> ( ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),V2f))
=> p(ap(V0P,V3s)) ) )
=> p(ap(V0P,ap(c_2Epred__set_2EBIGINTER(A_27a),V2f))) ) )
=> p(ap(V0P,ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s))) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__EQ,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2f: $i] :
( mem(V2f,arr(arr(A_27a,bool),bool))
=> ( ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),V2f))
=> p(ap(V0P,V3s)) ) )
=> p(ap(V0P,ap(c_2Epred__set_2EBIGINTER(A_27a),V2f))) ) )
=> ( ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s) = V1s )
<=> p(ap(V0P,V1s)) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__HULL,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s)) = ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__SUBSET,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s))) ) ) ).
tff(conj_thm_2Etopology_2EHULL__MONO,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s)),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2t))) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__ANTIMONO,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1Q: $i] :
( mem(V1Q,arr(arr(A_27a,bool),bool))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0P),V1Q))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Etopology_2Ehull(A_27a),V1Q),V2s)),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2s))) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__MINIMAL,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t))
& p(ap(V0P,V2t)) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s)),V2t)) ) ) ) ) ).
tff(conj_thm_2Etopology_2ESUBSET__HULL,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( p(ap(V0P,V2t))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s)),V2t))
<=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)) ) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__UNIQUE,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t))
& p(ap(V0P,V2t))
& ! [V3t_27: $i] :
( mem(V3t_27,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V3t_27))
& p(ap(V0P,V3t_27)) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V3t_27)) ) ) )
=> ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s) = V2t ) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__UNION__SUBSET,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s)),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2t))),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)))) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__UNION,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)) = ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s)),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2t))) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__UNION__LEFT,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)) = ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Epred__set_2EUNION(A_27a),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s)),V2t)) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__UNION__RIGHT,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)) = ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2t))) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__REDUNDANT__EQ,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1a),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2s)))
<=> ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1a),V2s)) = ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2s) ) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__REDUNDANT,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1a),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2s)))
=> ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1a),V2s)) = ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2s) ) ) ) ) ) ).
tff(lamtp_f2017,type,
f2017: ( del * $i ) > $i ).
tff(lameq_f2017,axiom,
! [A_27a: del,V1p: $i] :
( mem(V1p,arr(A_27a,bool))
=> ! [V4x: $i] : ( ap(f2017(A_27a,V1p),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(V1p,V4x)) ) ) ).
tff(conj_thm_2Etopology_2EHULL__INDUCT,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1p: $i] :
( mem(V1p,arr(A_27a,bool))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(V1p,V3x)) ) )
& p(ap(V0P,ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f2017(A_27a,V1p)))) )
=> ! [V5x: $i] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2s)))
=> p(ap(V1p,V5x)) ) ) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__INC,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s))) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__IMAGE__SUBSET,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(V0P,ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2s)))
& ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(V0P,V3s))
=> p(ap(V0P,ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V1f),V3s))) ) ) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V1f),V2s))),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V1f),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2s)))) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__IMAGE__GALOIS,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,A_27a))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( ( ! [V4s: $i] :
( mem(V4s,arr(A_27a,bool))
=> p(ap(V0P,ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V4s))) )
& ! [V5s: $i] :
( mem(V5s,arr(A_27a,bool))
=> ( p(ap(V0P,V5s))
=> p(ap(V0P,ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V1f),V5s))) ) )
& ! [V6s: $i] :
( mem(V6s,arr(A_27a,bool))
=> ( p(ap(V0P,V6s))
=> p(ap(V0P,ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V2g),V6s))) ) )
& ! [V7s: $i] :
( mem(V7s,arr(A_27a,bool))
=> ! [V8t: $i] :
( mem(V8t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V7s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V2g),V8t)))
<=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V1f),V7s)),V8t)) ) ) ) )
=> ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V1f),V3s)) = ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V1f),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V3s)) ) ) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__IMAGE,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> p(ap(V0P,ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V3s))) )
& ! [V4s: $i] :
( mem(V4s,arr(A_27a,bool))
=> ( p(ap(V0P,ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V1f),V4s)))
<=> p(ap(V0P,V4s)) ) )
& ! [V5x: $i] :
( mem(V5x,A_27a)
=> ! [V6y: $i] :
( mem(V6y,A_27a)
=> ( ( ap(V1f,V5x) = ap(V1f,V6y) )
=> ( V5x = V6y ) ) ) )
& ! [V7y: $i] :
( mem(V7y,A_27a)
=> ? [V8x: $i] :
( mem(V8x,A_27a)
& ( ap(V1f,V8x) = V7y ) ) ) )
=> ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V1f),V2s)) = ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V1f),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2s)) ) ) ) ) ) ).
tff(conj_thm_2Etopology_2EIS__HULL,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2f: $i] :
( mem(V2f,arr(arr(A_27a,bool),bool))
=> ( ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),V2f))
=> p(ap(V0P,V3s)) ) )
=> p(ap(V0P,ap(c_2Epred__set_2EBIGINTER(A_27a),V2f))) ) )
=> ( p(ap(V0P,V1s))
<=> ? [V4t: $i] :
( mem(V4t,arr(A_27a,bool))
& ( V1s = ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V4t) ) ) ) ) ) ) ).
tff(conj_thm_2Etopology_2EHULLS__EQ,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( ! [V3f: $i] :
( mem(V3f,arr(arr(A_27a,bool),bool))
=> ( ! [V4s: $i] :
( mem(V4s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4s),V3f))
=> p(ap(V0P,V4s)) ) )
=> p(ap(V0P,ap(c_2Epred__set_2EBIGINTER(A_27a),V3f))) ) )
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2t)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s))) )
=> ( ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s) = ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2t) ) ) ) ) ) ).
tff(lamtp_f2018,type,
f2018: ( del * $i * $i ) > $i ).
tff(lameq_f2018,axiom,
! [A_27a: del,V1P: $i] :
( mem(V1P,arr(arr(A_27a,bool),bool))
=> ! [V2Q: $i] :
( mem(V2Q,arr(arr(A_27a,bool),bool))
=> ! [V8x: $i] : ( ap(f2018(A_27a,V1P,V2Q),V8x) = ap(ap(c_2Ebool_2E_2F_5C,ap(V1P,V8x)),ap(V2Q,V8x)) ) ) ) ).
tff(conj_thm_2Etopology_2EHULL__P__AND__Q,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1P: $i] :
( mem(V1P,arr(arr(A_27a,bool),bool))
=> ! [V2Q: $i] :
( mem(V2Q,arr(arr(A_27a,bool),bool))
=> ( ( ! [V3f: $i] :
( mem(V3f,arr(arr(A_27a,bool),bool))
=> ( ! [V4s: $i] :
( mem(V4s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4s),V3f))
=> p(ap(V1P,V4s)) ) )
=> p(ap(V1P,ap(c_2Epred__set_2EBIGINTER(A_27a),V3f))) ) )
& ! [V5f: $i] :
( mem(V5f,arr(arr(A_27a,bool),bool))
=> ( ! [V6s: $i] :
( mem(V6s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V6s),V5f))
=> p(ap(V2Q,V6s)) ) )
=> p(ap(V2Q,ap(c_2Epred__set_2EBIGINTER(A_27a),V5f))) ) )
& ! [V7s: $i] :
( mem(V7s,arr(A_27a,bool))
=> ( p(ap(V2Q,V7s))
=> p(ap(V2Q,ap(ap(c_2Etopology_2Ehull(A_27a),V1P),V7s))) ) ) )
=> ( ap(ap(c_2Etopology_2Ehull(A_27a),f2018(A_27a,V1P,V2Q)),V0s) = ap(ap(c_2Etopology_2Ehull(A_27a),V1P),ap(ap(c_2Etopology_2Ehull(A_27a),V2Q),V0s)) ) ) ) ) ) ).
%------------------------------------------------------------------------------