ITP001 Axioms: ITP149^7.ax
%------------------------------------------------------------------------------
% File : ITP149^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : real_topology.ax [Gau19]
% : HL4149^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 1919 ( 307 unt; 157 typ; 0 def)
% Number of atoms : 10523 (1358 equ; 385 cnn)
% Maximal formula atoms : 38 ( 5 avg)
% Number of connectives : 37896 ( 385 ~; 106 |;2031 &;33163 @)
% ( 393 <=>;1818 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 10 avg;33163 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 4205 (4205 >; 0 *; 0 +; 0 <<)
% Number of symbols : 156 ( 154 usr; 10 con; 0-6 aty)
% Number of variables : 6746 ( 756 ^5481 !; 439 ?;6746 :)
% ( 70 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Ereal__topology_2Enet,type,
tyop_2Ereal__topology_2Enet: $tType > $tType ).
thf(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
thf(tyop_2Etopology_2Etopology,type,
tyop_2Etopology_2Etopology: $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ereal__topology_2E_2D_2D_3E,type,
c_2Ereal__topology_2E_2D_2D_3E:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > ( tyop_2Ereal__topology_2Enet @ A_27a ) > $o ) ).
thf(c_2Eiterate_2E_2E_2E,type,
c_2Eiterate_2E_2E_2E: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Ereal_2E_2F,type,
c_2Ereal_2E_2F: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Earithmetic_2E_3E_3D,type,
c_2Earithmetic_2E_3E_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Epred__set_2EBIGINTER,type,
c_2Epred__set_2EBIGINTER:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EBIGUNION,type,
c_2Epred__set_2EBIGUNION:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2ECARD,type,
c_2Epred__set_2ECARD:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Ereal__topology_2ECLOSED__interval,type,
c_2Ereal__topology_2ECLOSED__interval: ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epred__set_2ECROSS,type,
c_2Epred__set_2ECROSS:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Ereal__topology_2EClosed,type,
c_2Ereal__topology_2EClosed: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Epred__set_2EDELETE,type,
c_2Epred__set_2EDELETE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epred__set_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EDISJOINT,type,
c_2Epred__set_2EDISJOINT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ereal__topology_2EDist,type,
c_2Ereal__topology_2EDist: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Ecardinal_2EHAS__SIZE,type,
c_2Ecardinal_2EHAS__SIZE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum > $o ) ).
thf(c_2Elist_2EHD,type,
c_2Elist_2EHD:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a ) ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Earithmetic_2EMAX,type,
c_2Earithmetic_2EMAX: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ereal__topology_2EOPEN__interval,type,
c_2Ereal__topology_2EOPEN__interval: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal__topology_2EOpen,type,
c_2Ereal__topology_2EOpen: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eiterate_2ESum,type,
c_2Eiterate_2ESum:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ereal_2Eabs,type,
c_2Ereal_2Eabs: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Eat,type,
c_2Ereal__topology_2Eat: tyop_2Erealax_2Ereal > ( tyop_2Ereal__topology_2Enet @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ereal__topology_2Eat__infinity,type,
c_2Ereal__topology_2Eat__infinity: tyop_2Ereal__topology_2Enet @ tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Eat__neginfinity,type,
c_2Ereal__topology_2Eat__neginfinity: tyop_2Ereal__topology_2Enet @ tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Eat__posinfinity,type,
c_2Ereal__topology_2Eat__posinfinity: tyop_2Ereal__topology_2Enet @ tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Eball,type,
c_2Ereal__topology_2Eball: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal__topology_2Ebetween,type,
c_2Ereal__topology_2Ebetween: tyop_2Erealax_2Ereal > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Ereal__topology_2Ebilinear,type,
c_2Ereal__topology_2Ebilinear: ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Ereal__topology_2Ebounded__def,type,
c_2Ereal__topology_2Ebounded__def: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ecardinal_2Ecardeq,type,
c_2Ecardinal_2Ecardeq:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Ereal__topology_2Ecauchy,type,
c_2Ereal__topology_2Ecauchy: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Ereal__topology_2Ecball,type,
c_2Ereal__topology_2Ecball: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Etopology_2Eclosed__in,type,
c_2Etopology_2Eclosed__in:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ereal__topology_2Eclosed__segment,type,
c_2Ereal__topology_2Eclosed__segment: ( tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal__topology_2Eclosest__point,type,
c_2Ereal__topology_2Eclosest__point: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Eclosure,type,
c_2Ereal__topology_2Eclosure: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal__topology_2Ecollinear,type,
c_2Ereal__topology_2Ecollinear: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Ecompact,type,
c_2Ereal__topology_2Ecompact: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Ecomplete,type,
c_2Ereal__topology_2Ecomplete: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Ecomponents,type,
c_2Ereal__topology_2Ecomponents: ( tyop_2Erealax_2Ereal > $o ) > ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Econdensation__point__of,type,
c_2Ereal__topology_2Econdensation__point__of: tyop_2Erealax_2Ereal > ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Econnected,type,
c_2Ereal__topology_2Econnected: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Econnected__component,type,
c_2Ereal__topology_2Econnected__component: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal__topology_2Econtinuous,type,
c_2Ereal__topology_2Econtinuous:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Erealax_2Ereal ) > ( tyop_2Ereal__topology_2Enet @ A_27a ) > $o ) ).
thf(c_2Ereal__topology_2Econtinuous__on,type,
c_2Ereal__topology_2Econtinuous__on: ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Epred__set_2Ecountable,type,
c_2Epred__set_2Ecountable:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ereal__topology_2Edependent,type,
c_2Ereal__topology_2Edependent: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Ediameter,type,
c_2Ereal__topology_2Ediameter: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Edim,type,
c_2Ereal__topology_2Edim: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Enum_2Enum ).
thf(c_2Ereal__topology_2Eeuclidean,type,
c_2Ereal__topology_2Eeuclidean: tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Eeventually,type,
c_2Ereal__topology_2Eeventually:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Ereal__topology_2Enet @ A_27a ) > $o ) ).
thf(c_2Ereal__topology_2Efrom,type,
c_2Ereal__topology_2Efrom: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Ereal__topology_2Efrontier,type,
c_2Ereal__topology_2Efrontier: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal__topology_2Efsigma,type,
c_2Ereal__topology_2Efsigma: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Egdelta,type,
c_2Ereal__topology_2Egdelta: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Ehausdist,type,
c_2Ereal__topology_2Ehausdist: ( tyop_2Epair_2Eprod @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) ) > tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Ehomeomorphic,type,
c_2Ereal__topology_2Ehomeomorphic: ( tyop_2Erealax_2Ereal > $o ) > ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Ehomeomorphism,type,
c_2Ereal__topology_2Ehomeomorphism: ( tyop_2Epair_2Eprod @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) ) > ( tyop_2Epair_2Eprod @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) ) > $o ).
thf(c_2Etopology_2Ehull,type,
c_2Etopology_2Ehull:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Ereal__topology_2Ein__direction,type,
c_2Ereal__topology_2Ein__direction: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > ( tyop_2Ereal__topology_2Enet @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ereal__topology_2Eindependent,type,
c_2Ereal__topology_2Eindependent: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Eiterate_2Einf,type,
c_2Eiterate_2Einf: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Einfsum,type,
c_2Ereal__topology_2Einfsum: ( tyop_2Enum_2Enum > $o ) > ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Einterior,type,
c_2Ereal__topology_2Einterior: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Einv,type,
c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Eis__interval,type,
c_2Ereal__topology_2Eis__interval: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Eisnet,type,
c_2Ereal__topology_2Eisnet:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).
thf(c_2Etopology_2Eistopology,type,
c_2Etopology_2Eistopology:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > $o ) ).
thf(c_2Ereal__topology_2Elim__def,type,
c_2Ereal__topology_2Elim__def:
!>[A_27a: $tType] : ( ( tyop_2Ereal__topology_2Enet @ A_27a ) > ( A_27a > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Ereal__topology_2Elimit__point__of,type,
c_2Ereal__topology_2Elimit__point__of: tyop_2Erealax_2Ereal > ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Elinear,type,
c_2Ereal__topology_2Elinear: ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Ereal__topology_2Elocally,type,
c_2Ereal__topology_2Elocally: ( ( tyop_2Erealax_2Ereal > $o ) > $o ) > ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal_2Emax,type,
c_2Ereal_2Emax: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Emidpoint,type,
c_2Ereal__topology_2Emidpoint: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Emin,type,
c_2Ereal_2Emin: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Emk__net,type,
c_2Ereal__topology_2Emk__net:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( tyop_2Ereal__topology_2Enet @ A_27a ) ) ).
thf(c_2Ereal__topology_2Enetlimit,type,
c_2Ereal__topology_2Enetlimit:
!>[A_27a: $tType] : ( ( tyop_2Ereal__topology_2Enet @ A_27a ) > A_27a ) ).
thf(c_2Ereal__topology_2Enetord,type,
c_2Ereal__topology_2Enetord:
!>[A_27a: $tType] : ( ( tyop_2Ereal__topology_2Enet @ A_27a ) > A_27a > A_27a > $o ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Etopology_2Eopen__in,type,
c_2Etopology_2Eopen__in:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ereal__topology_2Eopen__segment,type,
c_2Ereal__topology_2Eopen__segment: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal__topology_2Epairwise,type,
c_2Ereal__topology_2Epairwise:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ereal__topology_2Epermutes,type,
c_2Ereal__topology_2Epermutes:
!>[A_27a: $tType] : ( ( A_27a > A_27a ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ereal_2Epow,type,
c_2Ereal_2Epow: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Eproduct_2Eproduct,type,
c_2Eproduct_2Eproduct:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__ge,type,
c_2Ereal_2Ereal__ge: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2Ereal__gt,type,
c_2Ereal_2Ereal__gt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Ereal__lt,type,
c_2Erealax_2Ereal__lt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2Ereal__lte,type,
c_2Ereal_2Ereal__lte: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Ereal__mul,type,
c_2Erealax_2Ereal__mul: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__neg,type,
c_2Erealax_2Ereal__neg: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__of__num,type,
c_2Ereal_2Ereal__of__num: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__sub,type,
c_2Ereal_2Ereal__sub: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Esequentially,type,
c_2Ereal__topology_2Esequentially: tyop_2Ereal__topology_2Enet @ tyop_2Enum_2Enum ).
thf(c_2Ereal__topology_2Esetdist,type,
c_2Ereal__topology_2Esetdist: ( tyop_2Epair_2Eprod @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) ) > tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Espan,type,
c_2Ereal__topology_2Espan: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal__topology_2Esphere,type,
c_2Ereal__topology_2Esphere: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal__topology_2Esubspace,type,
c_2Ereal__topology_2Esubspace: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Esubtopology,type,
c_2Ereal__topology_2Esubtopology:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > ( A_27a > $o ) > ( tyop_2Etopology_2Etopology @ A_27a ) ) ).
thf(c_2Ereal__topology_2Esummable,type,
c_2Ereal__topology_2Esummable: ( tyop_2Enum_2Enum > $o ) > ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Ereal__topology_2Esums,type,
c_2Ereal__topology_2Esums: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > ( tyop_2Enum_2Enum > $o ) > $o ).
thf(c_2Ereal_2Esup,type,
c_2Ereal_2Esup: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal ).
thf(c_2Etopology_2Etopology,type,
c_2Etopology_2Etopology:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > ( tyop_2Etopology_2Etopology @ A_27a ) ) ).
thf(c_2Etopology_2Etopspace,type,
c_2Etopology_2Etopspace:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > A_27a > $o ) ).
thf(c_2Ereal__topology_2Etrivial__limit,type,
c_2Ereal__topology_2Etrivial__limit:
!>[A_27a: $tType] : ( ( tyop_2Ereal__topology_2Enet @ A_27a ) > $o ) ).
thf(c_2Ereal__topology_2Euniformly__continuous__on,type,
c_2Ereal__topology_2Euniformly__continuous__on: ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Ewithin,type,
c_2Ereal__topology_2Ewithin:
!>[A_27a: $tType] : ( ( tyop_2Ereal__topology_2Enet @ A_27a ) > ( A_27a > $o ) > ( tyop_2Ereal__topology_2Enet @ A_27a ) ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Ereal__topology_2Efsigma,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Efsigma @ V0s )
<=> ? [V1g: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( c_2Epred__set_2Ecountable @ ( tyop_2Erealax_2Ereal > $o ) @ V1g )
& ! [V2c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2c @ V1g )
=> ( c_2Ereal__topology_2EClosed @ V2c ) )
& ( ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V1g )
= V0s ) ) ) ).
thf(thm_2Ereal__topology_2Egdelta,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Egdelta @ V0s )
<=> ? [V1g: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( c_2Epred__set_2Ecountable @ ( tyop_2Erealax_2Ereal > $o ) @ V1g )
& ! [V2u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2u @ V1g )
=> ( c_2Ereal__topology_2EOpen @ V2u ) )
& ( ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V1g )
= V0s ) ) ) ).
thf(thm_2Ereal__topology_2Elocally,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ V0P @ V1s )
<=> ! [V2w: tyop_2Erealax_2Ereal > $o,V3x: tyop_2Erealax_2Ereal] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V2w )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V2w ) )
=> ? [V4u: tyop_2Erealax_2Ereal > $o,V5v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V4u )
& ( V0P @ V5v )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V4u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4u @ V5v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5v @ V2w ) ) ) ) ).
thf(thm_2Ereal__topology_2Ehausdist,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal
@ ( c_2Ebool_2E_2F_5C
@ ( c_2Ebool_2E_7E
@ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o )
@ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) ) )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) )
@ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
@ ( c_2Ebool_2E_3F @ tyop_2Erealax_2Ereal
@ ^ [V4b: tyop_2Erealax_2Ereal] :
( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V5d: tyop_2Erealax_2Ereal] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5d
@ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V6x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V0s ) ) )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V7y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7y @ V1t ) ) ) ) )
@ ( c_2Ereal_2Ereal__lte @ V5d @ V4b ) ) ) ) )
@ ( c_2Ereal_2Esup
@ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V8x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V0s ) ) )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V9y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V9y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9y @ V1t ) ) ) ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2Esetdist,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ ( c_2Eiterate_2Einf
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V3y ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Eclosest__point,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosest__point @ V0s @ V1a )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] :
( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
@ ( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V3y: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V0s ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2x ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V3y ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Esummable,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esummable @ V0s @ V1f )
<=> ? [V2l: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esums @ V1f @ V2l @ V0s ) ) ).
thf(thm_2Ereal__topology_2Einfsum,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Einfsum @ V0s @ V1f )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V2l: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esums @ V1f @ V2l @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2Esums,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esums @ V0f @ V1l @ V2s )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V2s @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V3n ) ) @ V0f )
@ V1l
@ c_2Ereal__topology_2Esequentially ) ) ).
thf(thm_2Ereal__topology_2Ehomeomorphic,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V1t )
<=> ? [V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V2f @ V3g ) ) ) ).
thf(thm_2Ereal__topology_2Ehomeomorphism,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V2f @ V3g ) )
<=> ( ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
=> ( ( V3g @ ( V2f @ V4x ) )
= V4x ) )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2f @ V0s )
= V1t )
& ( c_2Ereal__topology_2Econtinuous__on @ V2f @ V0s )
& ! [V5y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V1t )
=> ( ( V2f @ ( V3g @ V5y ) )
= V5y ) )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3g @ V1t )
= V0s )
& ( c_2Ereal__topology_2Econtinuous__on @ V3g @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2Eis__interval,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eis__interval @ V0s )
<=> ! [V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal,V3x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2b @ V0s ) )
=> ( ( ( ( c_2Ereal_2Ereal__lte @ V1a @ V3x )
& ( c_2Ereal_2Ereal__lte @ V3x @ V2b ) )
| ( ( c_2Ereal_2Ereal__lte @ V2b @ V3x )
& ( c_2Ereal_2Ereal__lte @ V3x @ V1a ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2Emidpoint,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Erealax_2Ereal__add @ V0a @ V1b ) ) ) ).
thf(thm_2Ereal__topology_2Ebetween,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebetween @ V0x @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) )
<=> ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V0x ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Ecollinear,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecollinear @ V0s )
<=> ? [V1u: tyop_2Erealax_2Ereal] :
! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V0s ) )
=> ? [V4c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ V2x @ V3y )
= ( c_2Erealax_2Ereal__mul @ V4c @ V1u ) ) ) ) ).
thf(thm_2Ereal__topology_2Econdensation__point__of,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econdensation__point__of @ V0x @ V1s )
<=> ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V2t )
& ( c_2Ereal__topology_2EOpen @ V2t ) )
=> ( (~) @ ( c_2Epred__set_2Ecountable @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1s @ V2t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Enetlimit,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( c_2Ereal__topology_2Enetlimit @ A_27a @ V0net )
= ( c_2Emin_2E_40 @ A_27a
@ ^ [V1a: A_27a] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V2x: A_27a] : ( c_2Ebool_2E_7E @ ( c_2Ereal__topology_2Enetord @ A_27a @ V0net @ V2x @ V1a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Ebounded__def,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
<=> ? [V1a: tyop_2Erealax_2Ereal] :
! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V2x ) @ V1a ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__interval,axiom,
! [V0l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )] :
( ( c_2Ereal__topology_2ECLOSED__interval @ V0l )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Epair_2EFST @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Elist_2EHD @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ V0l ) ) @ V1x ) @ ( c_2Ereal_2Ereal__lte @ V1x @ ( c_2Epair_2ESND @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Elist_2EHD @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ V0l ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__interval,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ V0a @ V2x ) @ ( c_2Erealax_2Ereal__lt @ V2x @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Ediameter,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ediameter @ V0s )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ ( c_2Ereal_2Esup
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1x @ V2y ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Ecomponents,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecomponents @ V0s )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2Econnected__component,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x @ V2y )
<=> ? [V3t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3t @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V3t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V3t ) ) ) ).
thf(thm_2Ereal__topology_2Eball,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2y @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2y ) ) @ V1e ) ) ) ) ).
thf(thm_2Ereal__topology_2Ecball,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2y @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2y ) ) @ V1e ) ) ) ) ).
thf(thm_2Ereal__topology_2Esphere,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2y @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2y ) ) @ V1e ) ) ) ) ).
thf(thm_2Ereal__topology_2Eclosed__segment,axiom,
! [V0l: tyop_2Elist_2Elist @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )] :
( ( c_2Ereal__topology_2Eclosed__segment @ V0l )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1u: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1u ) @ ( c_2Epair_2EFST @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Elist_2EHD @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ V0l ) ) ) @ ( c_2Erealax_2Ereal__mul @ V1u @ ( c_2Epair_2ESND @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Elist_2EHD @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ V0l ) ) ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1u ) @ ( c_2Ereal_2Ereal__lte @ V1u @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Eopen__segment,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Econnected,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ( (~)
@ ? [V1e1: tyop_2Erealax_2Ereal > $o,V2e2: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V1e1 )
& ( c_2Ereal__topology_2EOpen @ V2e2 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 ) )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1e1 @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2e2 @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Elimit__point__of,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elimit__point__of @ V0x @ V1s )
<=> ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V2t )
& ( c_2Ereal__topology_2EOpen @ V2t ) )
=> ? [V3y: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V3y = V0x ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V2t ) ) ) ) ).
thf(thm_2Ereal__topology_2Einterior,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Einterior @ V0s )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x
@ ( c_2Ebool_2E_3F @ ( tyop_2Erealax_2Ereal > $o )
@ ^ [V2t: tyop_2Erealax_2Ereal > $o] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal__topology_2EOpen @ V2t ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V2t ) @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2t @ V0s ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Eclosure,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eclosure @ V0s )
= ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Efrontier,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Efrontier @ V0s )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ ( c_2Ereal__topology_2Einterior @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2Eisnet,axiom,
! [A_27a: $tType,A_27b: $tType,V0g: A_27a > A_27b > $o] :
( ( c_2Ereal__topology_2Eisnet @ A_27a @ A_27b @ V0g )
<=> ! [V1x: A_27b,V2y: A_27b] :
( ! [V3z: A_27a] :
( ( V0g @ V3z @ V1x )
=> ( V0g @ V3z @ V2y ) )
| ! [V4z: A_27a] :
( ( V0g @ V4z @ V2y )
=> ( V0g @ V4z @ V1x ) ) ) ) ).
thf(thm_2Ereal__topology_2Enet__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Ereal__topology_2Enet @ A_27a ) > A_27a > A_27a > $o] : ( c_2Ebool_2ETYPE__DEFINITION @ ( A_27a > A_27a > $o ) @ ( tyop_2Ereal__topology_2Enet @ A_27a ) @ ( c_2Ereal__topology_2Eisnet @ A_27a @ A_27a ) @ V0rep ) ).
thf(thm_2Ereal__topology_2Eat,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eat @ V0a )
= ( c_2Ereal__topology_2Emk__net @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V0a ) ) ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V0a ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2y @ V0a ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Eat__infinity,axiom,
( c_2Ereal__topology_2Eat__infinity
= ( c_2Ereal__topology_2Emk__net @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__ge @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) ) ) ).
thf(thm_2Ereal__topology_2Eat__posinfinity,axiom,
( c_2Ereal__topology_2Eat__posinfinity
= ( c_2Ereal__topology_2Emk__net @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__ge @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal__topology_2Eat__neginfinity,axiom,
( c_2Ereal__topology_2Eat__neginfinity
= ( c_2Ereal__topology_2Emk__net @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal__topology_2Esequentially,axiom,
( c_2Ereal__topology_2Esequentially
= ( c_2Ereal__topology_2Emk__net @ tyop_2Enum_2Enum
@ ^ [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_3D @ V0m @ V1n ) ) ) ).
thf(thm_2Ereal__topology_2Ewithin,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1s: A_27a > $o] :
( ( c_2Ereal__topology_2Ewithin @ A_27a @ V0net @ V1s )
= ( c_2Ereal__topology_2Emk__net @ A_27a
@ ^ [V2x: A_27a,V3y: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal__topology_2Enetord @ A_27a @ V0net @ V2x @ V3y ) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2Ein__direction,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1v: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ein__direction @ V0a @ V1v )
= ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0a )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2b: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2b
@ ( c_2Ebool_2E_3F @ tyop_2Erealax_2Ereal
@ ^ [V3c: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3c ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ V2b @ V0a ) @ ( c_2Erealax_2Ereal__mul @ V3c @ V1v ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Etrivial__limit,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net )
<=> ( ! [V1a: A_27a,V2b: A_27a] : ( V1a = V2b )
| ? [V3a: A_27a,V4b: A_27a] :
( ( (~) @ ( V3a = V4b ) )
& ! [V5x: A_27a] :
( ( (~) @ ( c_2Ereal__topology_2Enetord @ A_27a @ V0net @ V5x @ V3a ) )
& ( (~) @ ( c_2Ereal__topology_2Enetord @ A_27a @ V0net @ V5x @ V4b ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Eeventually,axiom,
! [A_27a: $tType,V0p: A_27a > $o,V1net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( c_2Ereal__topology_2Eeventually @ A_27a @ V0p @ V1net )
<=> ( ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V1net )
| ? [V2y: A_27a] :
( ? [V3x: A_27a] : ( c_2Ereal__topology_2Enetord @ A_27a @ V1net @ V3x @ V2y )
& ! [V4x: A_27a] :
( ( c_2Ereal__topology_2Enetord @ A_27a @ V1net @ V4x @ V2y )
=> ( V0p @ V4x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Etendsto,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V0f @ V1l @ V2net )
<=> ! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V4x: A_27a] : ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V1l ) ) @ V3e )
@ V2net ) ) ) ).
thf(thm_2Ereal__topology_2Elim__def,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elim__def @ A_27a @ V0net @ V1f )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V2l: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net ) ) ) ).
thf(thm_2Ereal__topology_2Efrom,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Efrom @ V0n )
= ( c_2Epred__set_2EGSPEC @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V1m: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ $o @ V1m @ ( c_2Earithmetic_2E_3C_3D @ V0n @ V1m ) ) ) ) ).
thf(thm_2Ereal__topology_2Eeuclidean,axiom,
( c_2Ereal__topology_2Eeuclidean
= ( c_2Etopology_2Etopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2EOpen ) ) ).
thf(thm_2Ereal__topology_2Eclosed__def,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
= ( c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2Eopen__def,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ? [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ! [V3x_27: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x_27 @ V1x ) ) @ V2e )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x_27 @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Esubtopology,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1u: A_27a > $o] :
( ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V1u )
= ( c_2Etopology_2Etopology @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V2s: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ ( c_2Epred__set_2EINTER @ A_27a @ V2s @ V1u ) @ ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V2s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Edim,axiom,
! [V0v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Edim @ V0v )
= ( c_2Emin_2E_40 @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] :
( c_2Ebool_2E_3F @ ( tyop_2Erealax_2Ereal > $o )
@ ^ [V2b: tyop_2Erealax_2Ereal > $o] : ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2b @ V0v ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal__topology_2Eindependent @ V2b ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0v @ ( c_2Ereal__topology_2Espan @ V2b ) ) @ ( c_2Ecardinal_2EHAS__SIZE @ tyop_2Erealax_2Ereal @ V2b @ V1n ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Eindependent,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eindependent @ V0s )
<=> ( (~) @ ( c_2Ereal__topology_2Edependent @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2Edependent,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Edependent @ V0s )
<=> ? [V1a: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EDELETE @ tyop_2Erealax_2Ereal @ V0s @ V1a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Espan,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Espan @ V0s )
= ( c_2Etopology_2Ehull @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Esubspace @ V0s ) ) ).
thf(thm_2Ereal__topology_2Esubspace,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esubspace @ V0s )
<=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0s )
& ! [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V1x @ V2y ) @ V0s ) )
& ! [V3c: tyop_2Erealax_2Ereal,V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V3c @ V4x ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2Ebilinear,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0f )
<=> ( ! [V1x: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2Elinear
@ ^ [V2y: tyop_2Erealax_2Ereal] : ( V0f @ V1x @ V2y ) )
& ! [V3y: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2Elinear
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( V0f @ V4x @ V3y ) ) ) ) ).
thf(thm_2Ereal__topology_2Elinear,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
<=> ( ! [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( V0f @ ( c_2Erealax_2Ereal__add @ V1x @ V2y ) )
= ( c_2Erealax_2Ereal__add @ ( V0f @ V1x ) @ ( V0f @ V2y ) ) )
& ! [V3c: tyop_2Erealax_2Ereal,V4x: tyop_2Erealax_2Ereal] :
( ( V0f @ ( c_2Erealax_2Ereal__mul @ V3c @ V4x ) )
= ( c_2Erealax_2Ereal__mul @ V3c @ ( V0f @ V4x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Edist,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) )
= ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal__topology_2Epermutes,axiom,
! [A_27a: $tType,V0p: A_27a > A_27a,V1s: A_27a > $o] :
( ( c_2Ereal__topology_2Epermutes @ A_27a @ V0p @ V1s )
<=> ( ! [V2x: A_27a] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) )
=> ( ( V0p @ V2x )
= V2x ) )
& ! [V3y: A_27a] :
( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V4x: A_27a] : ( c_2Emin_2E_3D @ A_27a @ ( V0p @ V4x ) @ V3y ) ) ) ) ).
thf(thm_2Ereal__topology_2Epairwise,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( c_2Ereal__topology_2Epairwise @ A_27a @ V0r @ V1s )
<=> ! [V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s )
& ( (~) @ ( V2x = V3y ) ) )
=> ( V0r @ V2x @ V3y ) ) ) ).
thf(thm_2Ereal__topology_2Euniformly__continuous__on,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V1s )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3d )
& ! [V4x: tyop_2Erealax_2Ereal,V5x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x_27 @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x_27 @ V4x ) ) @ V3d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V5x_27 ) @ ( V0f @ V4x ) ) ) @ V2e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Econtinuous__on,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
=> ! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4d )
& ! [V5x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x_27 @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x_27 @ V2x ) ) @ V4d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V5x_27 ) @ ( V0f @ V2x ) ) ) @ V3e ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Econtinuous,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V0f @ V1net )
= ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V0f @ ( V0f @ ( c_2Ereal__topology_2Enetlimit @ A_27a @ V1net ) ) @ V1net ) ) ).
thf(thm_2Ereal__topology_2Ecomplete,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecomplete @ V0s )
<=> ! [V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1f @ V2n ) @ V0s )
& ( c_2Ereal__topology_2Ecauchy @ V1f ) )
=> ? [V3l: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3l @ V0s )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V1f @ V3l @ c_2Ereal__topology_2Esequentially ) ) ) ) ).
thf(thm_2Ereal__topology_2Ecauchy,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecauchy @ V0s )
<=> ! [V1e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e )
=> ? [V2N: tyop_2Enum_2Enum] :
! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3E_3D @ V3m @ V2N )
& ( c_2Earithmetic_2E_3E_3D @ V4n @ V2N ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0s @ V3m ) @ ( V0s @ V4n ) ) ) @ V1e ) ) ) ) ).
thf(thm_2Ereal__topology_2Ecompact,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
<=> ! [V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1f @ V2n ) @ V0s )
=> ? [V3l: tyop_2Erealax_2Ereal,V4r: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3l @ V0s )
& ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V5m @ V6n )
=> ( c_2Eprim__rec_2E_3C @ ( V4r @ V5m ) @ ( V4r @ V6n ) ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ V1f @ V4r ) @ V3l @ c_2Ereal__topology_2Esequentially ) ) ) ) ).
thf(thm_2Ereal__topology_2EFSIGMA__COMPLEMENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Efsigma @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) )
= ( c_2Ereal__topology_2Egdelta @ V0s ) ) ).
thf(thm_2Ereal__topology_2EGDELTA__COMPLEMENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Egdelta @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) )
= ( c_2Ereal__topology_2Efsigma @ V0s ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__OPEN__MAP__IMAGE,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o,V1Q: ( tyop_2Erealax_2Ereal > $o ) > $o,V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V2f @ V3s )
& ! [V4t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3s ) @ V4t )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2f @ V3s ) ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2f @ V4t ) ) )
& ! [V5t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5t @ V3s )
& ( V0P @ V5t ) )
=> ( V1Q @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2f @ V5t ) ) )
& ( c_2Ereal__topology_2Elocally @ V0P @ V3s ) )
=> ( c_2Ereal__topology_2Elocally @ V1Q @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2f @ V3s ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o,V1Q: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ! [V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V2f )
& ! [V4x: tyop_2Erealax_2Ereal,V5y: tyop_2Erealax_2Ereal] :
( ( ( V2f @ V4x )
= ( V2f @ V5y ) )
=> ( V4x = V5y ) ) )
=> ( ( V0P @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2f @ V3s ) )
= ( V1Q @ V3s ) ) )
=> ! [V6f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V7s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V6f )
& ! [V8x: tyop_2Erealax_2Ereal,V9y: tyop_2Erealax_2Ereal] :
( ( ( V6f @ V8x )
= ( V6f @ V9y ) )
=> ( V8x = V9y ) ) )
=> ( ( c_2Ereal__topology_2Elocally @ V0P @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6f @ V7s ) )
= ( c_2Ereal__topology_2Elocally @ V1Q @ V7s ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__TRANSLATION,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ! [V1a: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( V0P
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V1a @ V3x )
@ V2s ) )
= ( V0P @ V2s ) )
=> ! [V4a: tyop_2Erealax_2Ereal,V5s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ V0P
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V4a @ V6x )
@ V5s ) )
= ( c_2Ereal__topology_2Elocally @ V0P @ V5s ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__LOCALLY,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o,V1Q: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ! [V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V2s @ V3t )
=> ( ( V0P @ V2s )
= ( V1Q @ V3t ) ) )
=> ! [V4s: tyop_2Erealax_2Ereal > $o,V5t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V4s @ V5t )
=> ( ( c_2Ereal__topology_2Elocally @ V0P @ V4s )
= ( c_2Ereal__topology_2Elocally @ V1Q @ V5t ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__LOCALLY,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o,V1Q: ( tyop_2Erealax_2Ereal > $o ) > $o,V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ! [V4s: tyop_2Erealax_2Ereal > $o,V5t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V4s @ V5t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V2f @ V3g ) )
=> ( ( V0P @ V4s )
= ( V1Q @ V5t ) ) )
=> ! [V6s: tyop_2Erealax_2Ereal > $o,V7t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V6s @ V7t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V2f @ V3g ) )
=> ( ( c_2Ereal__topology_2Elocally @ V0P @ V6s )
= ( c_2Ereal__topology_2Elocally @ V1Q @ V7t ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__INTER,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ! [V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( V0P @ V1s )
& ( V0P @ V2t ) )
=> ( V0P @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1s @ V2t ) ) )
=> ! [V3s: tyop_2Erealax_2Ereal > $o,V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elocally @ V0P @ V3s )
& ( c_2Ereal__topology_2Elocally @ V0P @ V4t ) )
=> ( c_2Ereal__topology_2Elocally @ V0P @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V3s @ V4t ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__SING,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elocally @ V0P @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
= ( V0P @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__EMPTY,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o] : ( c_2Ereal__topology_2Elocally @ V0P @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__DIFF__CLOSED,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elocally @ V0P @ V1s )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V2t ) )
=> ( c_2Ereal__topology_2Elocally @ V0P @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V1s @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__OPEN__SUBSET,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elocally @ V0P @ V1s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V2t ) )
=> ( c_2Ereal__topology_2Elocally @ V0P @ V2t ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__MONO,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o,V1Q: ( tyop_2Erealax_2Ereal > $o ) > $o,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ! [V3t: tyop_2Erealax_2Ereal > $o] :
( ( V0P @ V3t )
=> ( V1Q @ V3t ) )
& ( c_2Ereal__topology_2Elocally @ V0P @ V2s ) )
=> ( c_2Ereal__topology_2Elocally @ V1Q @ V2s ) ) ).
thf(thm_2Ereal__topology_2EURYSOHN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ? [V4f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V4f @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
& ! [V5x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V4f @ V5x ) @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V6x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V0s )
=> ( ( V4f @ V6x )
= V2a ) )
& ! [V7x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V1t )
=> ( ( V4f @ V7x )
= V3b ) ) ) ) ).
thf(thm_2Ereal__topology_2EURYSOHN__STRONG,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( (~) @ ( V2a = V3b ) ) )
=> ? [V4f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V4f @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
& ! [V5x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V4f @ V5x ) @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V6x: tyop_2Erealax_2Ereal] :
( ( ( V4f @ V6x )
= V2a )
<=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V0s ) )
& ! [V7x: tyop_2Erealax_2Ereal] :
( ( ( V4f @ V7x )
= V3b )
<=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2EURYSOHN__LOCAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o,V3a: tyop_2Erealax_2Ereal,V4b: tyop_2Erealax_2Ereal] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V0s )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V1t )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ? [V5f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V5f @ V2u )
& ! [V6x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V2u )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V5f @ V6x ) @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V4b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) )
& ! [V7x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V0s )
=> ( ( V5f @ V7x )
= V3a ) )
& ! [V8x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V1t )
=> ( ( V5f @ V8x )
= V4b ) ) ) ) ).
thf(thm_2Ereal__topology_2EURYSOHN__LOCAL__STRONG,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o,V3a: tyop_2Erealax_2Ereal,V4b: tyop_2Erealax_2Ereal] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V0s )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V1t )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( (~) @ ( V3a = V4b ) ) )
=> ? [V5f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V5f @ V2u )
& ! [V6x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V2u )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V5f @ V6x ) @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V4b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) )
& ! [V7x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V2u )
=> ( ( ( V5f @ V7x )
= V3a )
<=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V0s ) ) )
& ! [V8x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V2u )
=> ( ( ( V5f @ V8x )
= V4b )
<=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V1t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EISOMETRY__IMP__HOMEOMORPHISM__COMPACT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V1s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V1s )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1s ) )
=> ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V2x ) @ ( V0f @ V3y ) ) )
= ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V3y ) ) ) ) )
=> ? [V4g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V1s ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V4g ) ) ) ).
thf(thm_2Ereal__topology_2EISOMETRY__IMP__EMBEDDING,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1s ) )
=> ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ ( V0f @ V4y ) ) )
= ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x @ V4y ) ) ) ) )
=> ? [V5g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V2t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V5g ) ) ) ).
thf(thm_2Ereal__topology_2EISOMETRY__IMP__OPEN__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3u: tyop_2Erealax_2Ereal > $o] :
( ( ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V4x: tyop_2Erealax_2Ereal,V5y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V1s ) )
=> ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ ( V0f @ V5y ) ) )
= ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4x @ V5y ) ) ) )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3u ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3u ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__DIAMETER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1e: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e ) )
=> ? [V2d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2d )
& ! [V3t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V3t )
& ( (~)
@ ( V3t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V3t ) ) @ V2d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal__topology_2Ediameter @ V0s ) @ ( c_2Ereal__topology_2Ediameter @ V3t ) ) ) @ V1e ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__ALT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal_2Esup
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__COMPACT__NONTRIVIAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Eiterate_2Einf
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2e: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2e
@ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
@ ( c_2Ebool_2E_2F_5C
@ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V3x @ V4y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1t ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V4y ) @ V2e ) ) ) ) ) )
@ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V5x: tyop_2Erealax_2Ereal,V6y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V5x @ V6y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V0s ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V6y ) @ V2e ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__EQ__0,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t ) )
=> ( ( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( ( c_2Ereal__topology_2Eclosure @ V0s )
= ( c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__TRANS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t )
& ( c_2Ereal__topology_2Ebounded__def @ V2u )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V2u ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V2u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__COMPACT__SUMS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2y: tyop_2Erealax_2Ereal,V3z: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V2y @ V3z ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V1t ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3z @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__TRIANGLE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t )
& ( c_2Ereal__topology_2Ebounded__def @ V2u )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V2u ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V2u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__COMPACT__EXISTS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ? [V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V3y ) ) @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__INSERT__LE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2a @ V0s ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2a @ V1t ) ) ) @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__UNION__LE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t )
& ( c_2Ereal__topology_2Ebounded__def @ V2u )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V2u
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V2u ) ) ) @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V2u ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__HAUSDIST__LE__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2b: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) @ V2b )
<=> ( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) @ V2b ) )
& ! [V4y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1t )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V4y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) @ V2b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__NONTRIVIAL__ALT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal_2Emax
@ ( c_2Ereal_2Esup
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) ) ) )
@ ( c_2Ereal_2Esup
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__NONTRIVIAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal_2Esup
@ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) ) )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EUPPER__LOWER__HEMICONTINUOUS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V2s )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V1t ) )
& ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V4u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V2s ) @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V4u ) ) ) ) ) )
& ! [V6u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V6u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V2s ) @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ V6u ) ) ) ) ) ) )
=> ! [V8x: tyop_2Erealax_2Ereal,V9e: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V2s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V9e )
& ( c_2Ereal__topology_2Ebounded__def @ ( V0f @ V8x ) ) )
=> ? [V10d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V10d )
& ! [V11x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V11x_27 @ V2s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8x @ V11x_27 ) ) @ V10d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( V0f @ V8x ) @ ( V0f @ V11x_27 ) ) ) @ V9e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LT__HAUSDIST__POINT__EXISTS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) @ V3d )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) )
=> ? [V4y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1t )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V4y ) ) @ V3d ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__SETDIST__TRIANGLE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V2u ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V2u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__HAUSDIST__TRIANGLE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Ereal__topology_2Ebounded__def @ V1t )
& ( c_2Ereal__topology_2Ebounded__def @ V2u ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V2u ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V2u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__SING__LE__HAUSDIST,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__LE__HAUSDIST,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LE__HAUSDIST,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal,V4c: tyop_2Erealax_2Ereal,V5z: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V6x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V6x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) @ V3b ) )
& ! [V7y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7y @ V1t )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V7y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) @ V4c ) )
& ( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5z @ V0s )
& ( c_2Ereal_2Ereal__lte @ V2a @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V5z @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) ) )
| ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5z @ V1t )
& ( c_2Ereal_2Ereal__lte @ V2a @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V5z @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ V2a @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__HAUSDIST__LE__SUMS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2b: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V3y: tyop_2Erealax_2Ereal,V4z: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V3y @ V4z ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4z @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2b ) ) ) ) ) ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V5y: tyop_2Erealax_2Ereal,V6z: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V5y @ V6z ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6z @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2b ) ) ) ) ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) @ V2b ) ) ).
thf(thm_2Ereal__topology_2EREAL__HAUSDIST__LE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2b: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) @ V2b ) )
& ! [V4y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1t )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V4y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) @ V2b ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) @ V2b ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__CLOSURE,axiom,
( ! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ V1t ) )
= ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) )
& ! [V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ ( c_2Ereal__topology_2Eclosure @ V3t ) ) )
= ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V3t ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( V0f @ V3x ) )
= ( c_2Ereal_2Eabs @ V3x ) ) )
=> ( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2t ) ) )
= ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V2t ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__TRANSLATION,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehausdist
@ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o )
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V3x )
@ V1s )
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V4x )
@ V2t ) ) )
= ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2s_27: tyop_2Erealax_2Ereal > $o,V3t_27: tyop_2Erealax_2Ereal > $o] :
( ! [V4b: tyop_2Erealax_2Ereal] :
( ( ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V5x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) @ V4b ) )
& ! [V6y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6y @ V1t )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V6y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) @ V4b ) ) )
<=> ( ! [V7x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V2s_27 )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V7x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V3t_27 ) ) @ V4b ) )
& ! [V8y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8y @ V3t_27 )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V8y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V2s_27 ) ) @ V4b ) ) ) )
=> ( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s_27 @ V3t_27 ) ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__SINGS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__EMPTY,axiom,
( ! [V0t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) @ V0t ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__SYM,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__REFL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V0s ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal__topology_2EHAUSDIST__POS__LE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2Ehausdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__NEST__GEN,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > $o] :
( ( ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2EClosed @ ( V0s @ V1n ) )
& ( c_2Ereal__topology_2Econnected @ ( V0s @ V1n ) ) )
& ? [V2n: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2Ecompact @ ( V0s @ V2n ) )
& ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V3m @ V4n )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0s @ V4n ) @ ( V0s @ V3m ) ) ) )
=> ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ tyop_2Enum_2Enum
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( V0s @ V5n ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__NEST,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > $o] :
( ( ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Ecompact @ ( V0s @ V1n ) )
& ( c_2Ereal__topology_2Econnected @ ( V0s @ V1n ) ) )
& ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V2m @ V3n )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0s @ V3n ) @ ( V0s @ V2m ) ) ) )
=> ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ tyop_2Enum_2Enum
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( V0s @ V4n ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__CHAIN__GEN,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0f )
=> ( ( c_2Ereal__topology_2EClosed @ V1s )
& ( c_2Ereal__topology_2Econnected @ V1s ) ) )
& ? [V2s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V0f )
& ( c_2Ereal__topology_2Ecompact @ V2s ) )
& ! [V3s: tyop_2Erealax_2Ereal > $o,V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V3s @ V0f )
& ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V4t @ V0f ) )
=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3s @ V4t )
| ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4t @ V3s ) ) ) )
=> ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__CHAIN,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0f )
=> ( ( c_2Ereal__topology_2Ecompact @ V1s )
& ( c_2Ereal__topology_2Econnected @ V1s ) ) )
& ! [V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V0f )
& ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V3t @ V0f ) )
=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2s @ V3t )
| ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3t @ V2s ) ) ) )
=> ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2ESEPARATION__T0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V0x = V1y ) )
<=> ? [V2u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V2u )
& ( (~)
@ ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V2u )
= ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ V2u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEPARATION__T1,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V0x = V1y ) )
<=> ? [V2u: tyop_2Erealax_2Ereal > $o,V3v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V2u )
& ( c_2Ereal__topology_2EOpen @ V3v )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V2u )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ V2u ) )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V3v ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ V3v ) ) ) ).
thf(thm_2Ereal__topology_2ESEPARATION__T2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V0x = V1y ) )
<=> ? [V2u: tyop_2Erealax_2Ereal > $o,V3v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V2u )
& ( c_2Ereal__topology_2EOpen @ V3v )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V2u )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ V3v )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2u @ V3v )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEPARATION__HAUSDORFF,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V0x = V1y ) )
=> ? [V2u: tyop_2Erealax_2Ereal > $o,V3v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V2u )
& ( c_2Ereal__topology_2EOpen @ V3v )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V2u )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ V3v )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2u @ V3v )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEPARATION__NORMAL__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ? [V2u: tyop_2Erealax_2Ereal > $o,V3v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V2u )
& ( c_2Ereal__topology_2Ecompact @ ( c_2Ereal__topology_2Eclosure @ V2u ) )
& ( c_2Ereal__topology_2EOpen @ V3v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V2u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V3v )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2u @ V3v )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEPARATION__NORMAL__LOCAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V0s )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V1t )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ? [V3s_27: tyop_2Erealax_2Ereal > $o,V4t_27: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V3s_27 )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V4t_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V3s_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V4t_27 )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V3s_27 @ V4t_27 )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEPARATION__NORMAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ? [V2u: tyop_2Erealax_2Ereal > $o,V3v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V2u )
& ( c_2Ereal__topology_2EOpen @ V3v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V2u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V3v )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2u @ V3v )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEPARATION__CLOSURES,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Eclosure @ V1t ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1t @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ? [V2u: tyop_2Erealax_2Ereal > $o,V3v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EDISJOINT @ tyop_2Erealax_2Ereal @ V2u @ V3v )
& ( c_2Ereal__topology_2EOpen @ V2u )
& ( c_2Ereal__topology_2EOpen @ V3v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V2u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V3v ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__BALLS,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2r ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V3s ) ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2E_5C_2F @ ( c_2Ereal_2Ereal__lte @ V2r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2Ereal__lte @ V3s @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Emax @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Erealax_2Ereal__add @ V2r @ V3s ) ) ) ) )
& ! [V4a: tyop_2Erealax_2Ereal,V5b: tyop_2Erealax_2Ereal,V6r: tyop_2Erealax_2Ereal,V7s: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V6r ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5b @ V7s ) ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2E_5C_2F @ ( c_2Ereal_2Ereal__lte @ V6r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Erealax_2Ereal__lt @ V7s @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Emax @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V5b ) ) @ ( c_2Erealax_2Ereal__add @ V6r @ V7s ) ) ) ) )
& ! [V8a: tyop_2Erealax_2Ereal,V9b: tyop_2Erealax_2Ereal,V10r: tyop_2Erealax_2Ereal,V11s: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V10r ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V9b @ V11s ) ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2E_5C_2F @ ( c_2Erealax_2Ereal__lt @ V10r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2Ereal__lte @ V11s @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Emax @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V9b ) ) @ ( c_2Erealax_2Ereal__add @ V10r @ V11s ) ) ) ) )
& ! [V12a: tyop_2Erealax_2Ereal,V13b: tyop_2Erealax_2Ereal,V14r: tyop_2Erealax_2Ereal,V15s: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V12a @ V14r ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V13b @ V15s ) ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2E_5C_2F @ ( c_2Erealax_2Ereal__lt @ V14r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Erealax_2Ereal__lt @ V15s @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Emax @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V12a @ V13b ) ) @ ( c_2Erealax_2Ereal__add @ V14r @ V15s ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__LE__SING,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__SING__TRIANGLE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2y ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__SING__FRONTIER__CASES,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__SING__IN__SET,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V1s )
=> ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1s ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED__IN,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0u ) )
=> ( ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1s ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EClosed @ V0s )
=> ( ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__EQ__0__SING,axiom,
( ! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) ) )
& ! [V2s: tyop_2Erealax_2Ereal > $o,V3x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ( V2s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Ereal__topology_2Eclosure @ V2s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__CLOSEST__POINT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V1s ) )
= ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal__topology_2Eclosest__point @ V1s @ V0a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__SING__FRONTIER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) )
=> ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) )
= ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__FRONTIERS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EDISJOINT @ tyop_2Erealax_2Ereal @ V0s @ V1t ) @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2Efrontier @ V0s ) @ ( c_2Ereal__topology_2Efrontier @ V1t ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__ZERO__STRONG,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( (~) @ ( c_2Epred__set_2EDISJOINT @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) )
=> ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__ZERO,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( (~) @ ( c_2Epred__set_2EDISJOINT @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
=> ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__UNIV,axiom,
( ! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V1t ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__UNIQUE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal,V4d: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2a @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3b @ V1t )
& ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) )
= V4d )
& ! [V5x: tyop_2Erealax_2Ereal,V6y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6y @ V1t ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x @ V6y ) ) ) ) )
=> ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= V4d ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( V0f @ V3x ) )
= ( c_2Ereal_2Eabs @ V3x ) ) )
=> ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2t ) ) )
= ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V2t ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__TRANSLATION,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esetdist
@ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o )
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V3x )
@ V1s )
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V4x )
@ V2t ) ) )
= ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__EQ__0__BOUNDED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
| ( c_2Ereal__topology_2Ebounded__def @ V1t ) )
=> ( ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ ( c_2Ereal__topology_2Eclosure @ V1t ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t ) )
=> ( ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__EQ__0__COMPACT__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t ) )
=> ( ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__CLOSED__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t )
& ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V3y ) )
= ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__COMPACT__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t )
& ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V3y ) )
= ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__FRONTIER,axiom,
( ! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EDISJOINT @ tyop_2Erealax_2Ereal @ V0s @ V1t )
=> ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2Efrontier @ V0s ) @ V1t ) )
= ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) )
& ! [V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EDISJOINT @ tyop_2Erealax_2Ereal @ V2s @ V3t )
=> ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ ( c_2Ereal__topology_2Efrontier @ V3t ) ) )
= ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V3t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__CLOSURE,axiom,
( ! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ V1t ) )
= ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) )
& ! [V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ ( c_2Ereal__topology_2Eclosure @ V3t ) ) )
= ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V3t ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__SUBSET__LEFT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V2u ) ) @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V2u ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__SUBSET__RIGHT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V2u ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V2u ) ) @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__DIFFERENCES,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal__topology_2Esetdist
@ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal_2Ereal__sub @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SETDIST,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( c_2Ereal__topology_2Euniformly__continuous__on
@ ^ [V2y: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) )
@ V1t ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__SETDIST,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( c_2Ereal__topology_2Econtinuous__on
@ ^ [V2y: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) )
@ V1t ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__SETDIST,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal
@ ^ [V2y: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) )
@ ( c_2Ereal__topology_2Eat @ V1x ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__LIPSCHITZ,axiom,
! [A_27a: $tType,V0s: tyop_2Erealax_2Ereal > $o,V1t: A_27a,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0s ) ) ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V3y ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__SINGS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__TRIANGLE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2t: tyop_2Erealax_2Ereal > $o] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V2t ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V2t ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__SYM,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) )
= ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__REFL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V0s ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal__topology_2EREAL__SETDIST__LT__EXISTS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2b: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) @ V2b ) )
=> ? [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1t )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x @ V4y ) ) @ V2b ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LE__SETDIST__EQ,axiom,
! [V0d: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal_2Ereal__lte @ V0d @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V2t ) ) )
<=> ( ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V2t ) )
=> ( c_2Ereal_2Ereal__lte @ V0d @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x @ V4y ) ) ) )
& ( ( ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( V2t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ( c_2Ereal_2Ereal__lte @ V0d @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__LE__DIST,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V3y ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LE__SETDIST,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2d: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1t ) )
=> ( c_2Ereal_2Ereal__lte @ V2d @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x @ V4y ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ V2d @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__SUBSETS__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2s_27: tyop_2Erealax_2Ereal > $o,V3t_27: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2s_27 @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3t_27 @ V1t )
& ! [V4x: tyop_2Erealax_2Ereal,V5y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V1t ) )
=> ? [V6x_27: tyop_2Erealax_2Ereal,V7y_27: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x_27 @ V2s_27 )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7y_27 @ V3t_27 )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6x_27 @ V7y_27 ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4x @ V5y ) ) ) ) ) )
=> ( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s_27 @ V3t_27 ) )
= ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__POS__LE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ESETDIST__EMPTY,axiom,
( ! [V0t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) @ V0t ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esetdist @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSEST__POINT__IN__FRONTIER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2Einterior @ V0s ) ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosest__point @ V0s @ V1x ) @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSEST__POINT__IN__INTERIOR,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosest__point @ V0s @ V1x ) @ ( c_2Ereal__topology_2Einterior @ V0s ) )
= ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEGMENT__TO__POINT__EXISTS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V2b: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2b @ V0s )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEGMENT__TO__CLOSEST__POINT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Ereal__topology_2Eclosest__point @ V0s @ V1a ) ) ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__DIST__CLOSEST__POINT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ereal__topology_2Euniformly__continuous__on
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosest__point @ V0s @ V2x ) ) )
@ V1t ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__DIST__CLOSEST__POINT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosest__point @ V0s @ V2x ) ) )
@ V1t ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__DIST__CLOSEST__POINT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosest__point @ V0s @ V2x ) ) )
@ ( c_2Ereal__topology_2Eat @ V1x ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__CLOSEST__POINT__LIPSCHITZ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2Eclosest__point @ V0s @ V1x ) ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2y @ ( c_2Ereal__topology_2Eclosest__point @ V0s @ V2y ) ) ) ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2y ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSEST__POINT__REFL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( ( c_2Ereal__topology_2Eclosest__point @ V0s @ V1x )
= V1x )
<=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSEST__POINT__SELF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ( ( c_2Ereal__topology_2Eclosest__point @ V0s @ V1x )
= V1x ) ) ).
thf(thm_2Ereal__topology_2ECLOSEST__POINT__LE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Ereal__topology_2Eclosest__point @ V0s @ V1a ) ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSEST__POINT__IN__SET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosest__point @ V0s @ V1a ) @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSEST__POINT__EXISTS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosest__point @ V0s @ V1a ) @ V0s )
& ! [V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Ereal__topology_2Eclosest__point @ V0s @ V1a ) ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2y ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDINI,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V2s )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2Econtinuous__on @ ( V0f @ V3n ) @ V2s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V2s )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V2s )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V5n: tyop_2Enum_2Enum] : ( V0f @ V5n @ V4x )
@ ( V1g @ V4x )
@ c_2Ereal__topology_2Esequentially ) )
& ! [V6n: tyop_2Enum_2Enum,V7x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V2s )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V6n @ V7x ) @ ( V0f @ ( c_2Earithmetic_2E_2B @ V6n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V7x ) ) ) )
=> ! [V8e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V8e )
=> ( c_2Ereal__topology_2Eeventually @ tyop_2Enum_2Enum
@ ^ [V9n: tyop_2Enum_2Enum] :
( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V10x: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10x @ V2s ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V0f @ V9n @ V10x ) @ ( V1g @ V10x ) ) ) @ V8e ) ) )
@ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2EBANACH__FIX,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecomplete @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2c )
& ( c_2Erealax_2Ereal__lt @ V2c @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V1s )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1s ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ ( V0f @ V4y ) ) ) @ ( c_2Erealax_2Ereal__mul @ V2c @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x @ V4y ) ) ) ) ) )
=> ( c_2Ebool_2E_3F_21 @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V5x ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__REARRANGE,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2p: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( ( c_2Ereal__topology_2Esummable @ V1s
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0x @ V3n ) ) )
& ( c_2Ereal__topology_2Epermutes @ tyop_2Enum_2Enum @ V2p @ V1s ) )
=> ( c_2Ereal__topology_2Esummable @ V1s @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ V0x @ V2p ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__REARRANGE,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2p: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V3l: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Esummable @ V1s
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0x @ V4n ) ) )
& ( c_2Ereal__topology_2Epermutes @ tyop_2Enum_2Enum @ V2p @ V1s )
& ( c_2Ereal__topology_2Esums @ V0x @ V3l @ V1s ) )
=> ( c_2Ereal__topology_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ V0x @ V2p ) @ V3l @ V1s ) ) ).
thf(thm_2Ereal__topology_2ESERIES__REARRANGE__EQ,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2p: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V3l: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Esummable @ V1s
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0x @ V4n ) ) )
& ( c_2Ereal__topology_2Epermutes @ tyop_2Enum_2Enum @ V2p @ V1s ) )
=> ( ( c_2Ereal__topology_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ V0x @ V2p ) @ V3l @ V1s )
= ( c_2Ereal__topology_2Esums @ V0x @ V3l @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__INJECTIVE__IMAGE,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V3l: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Esummable @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2f @ V1s )
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0x @ V4n ) ) )
& ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
( ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5m @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V6n @ V1s )
& ( ( V2f @ V5m )
= ( V2f @ V6n ) ) )
=> ( V5m = V6n ) ) )
=> ( ( c_2Ereal__topology_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ V0x @ V2f ) @ V3l @ V1s )
= ( c_2Ereal__topology_2Esums @ V0x @ V3l @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2f @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__INJECTIVE__IMAGE__STRONG,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( ( c_2Ereal__topology_2Esummable @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2f @ V1s )
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0x @ V3n ) ) )
& ! [V4m: tyop_2Enum_2Enum,V5n: tyop_2Enum_2Enum] :
( ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4m @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5n @ V1s )
& ( ( V2f @ V4m )
= ( V2f @ V5n ) ) )
=> ( V4m = V5n ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V6n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2f @ V1s ) @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V6n ) ) @ V0x ) @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V1s @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V6n ) ) @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ V0x @ V2f ) ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ c_2Ereal__topology_2Esequentially ) ) ).
thf(thm_2Ereal__topology_2ESERIES__DIRICHLET,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2N: tyop_2Enum_2Enum,V3k: tyop_2Enum_2Enum,V4m: tyop_2Enum_2Enum] :
( ( ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V4m @ V5n ) @ V0f ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
& ! [V6n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V2N @ V6n )
=> ( c_2Ereal_2Ereal__lte @ ( V1g @ ( c_2Earithmetic_2E_2B @ V6n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V1g @ V6n ) ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V1g @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ c_2Ereal__topology_2Esequentially ) )
=> ( c_2Ereal__topology_2Esummable @ ( c_2Ereal__topology_2Efrom @ V3k )
@ ^ [V7n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V1g @ V7n ) @ ( V0f @ V7n ) ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__DIRICHLET__BILINEAR,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3k: tyop_2Enum_2Enum,V4m: tyop_2Enum_2Enum,V5p: tyop_2Enum_2Enum,V6l: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebilinear @ V2h )
& ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum
@ ^ [V7n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V4m @ V7n ) @ V0f ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V7n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
& ( c_2Ereal__topology_2Esummable @ ( c_2Ereal__topology_2Efrom @ V5p )
@ ^ [V8n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V1g @ ( c_2Earithmetic_2E_2B @ V8n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V1g @ V8n ) ) ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V9n: tyop_2Enum_2Enum] : ( V2h @ ( V1g @ ( c_2Earithmetic_2E_2B @ V9n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V9n ) @ V0f ) )
@ V6l
@ c_2Ereal__topology_2Esequentially ) )
=> ( c_2Ereal__topology_2Esummable @ ( c_2Ereal__topology_2Efrom @ V3k )
@ ^ [V10n: tyop_2Enum_2Enum] : ( V2h @ ( V1g @ V10n ) @ ( V0f @ V10n ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__BILINEAR__PARTIAL__PRE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal,V4k: tyop_2Enum_2Enum] :
( ( ( c_2Ereal__topology_2Ebilinear @ V2h )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V5n: tyop_2Enum_2Enum] : ( V2h @ ( V0f @ ( c_2Earithmetic_2E_2B @ V5n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V1g @ V5n ) )
@ V3l
@ c_2Ereal__topology_2Esequentially )
& ( c_2Ereal__topology_2Esummable @ ( c_2Ereal__topology_2Efrom @ V4k )
@ ^ [V6n: tyop_2Enum_2Enum] : ( V2h @ ( c_2Ereal_2Ereal__sub @ ( V0f @ ( c_2Earithmetic_2E_2B @ V6n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V0f @ V6n ) ) @ ( V1g @ V6n ) ) ) )
=> ( c_2Ereal__topology_2Esummable @ ( c_2Ereal__topology_2Efrom @ V4k )
@ ^ [V7n: tyop_2Enum_2Enum] : ( V2h @ ( V0f @ V7n ) @ ( c_2Ereal_2Ereal__sub @ ( V1g @ V7n ) @ ( V1g @ ( c_2Earithmetic_2E_2D @ V7n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__PARTIAL__SUMS,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1k @ V2n ) @ V0f ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
=> ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )
@ ( c_2Epair_2EUNCURRY @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V3m @ V4n ) @ V0f ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3m @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__RATIO,axiom,
! [V0c: tyop_2Erealax_2Ereal,V1a: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o,V3N: tyop_2Enum_2Enum] :
( ( ( c_2Erealax_2Ereal__lt @ V0c @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V4n @ V3N )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V1a @ ( c_2Enum_2ESUC @ V4n ) ) ) @ ( c_2Erealax_2Ereal__mul @ V0c @ ( c_2Ereal_2Eabs @ ( V1a @ V4n ) ) ) ) ) )
=> ? [V5l: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esums @ V1a @ V5l @ V2s ) ) ).
thf(thm_2Ereal__topology_2ESERIES__COMPARISON__UNIFORM,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2P: A_27a > $o,V3s: tyop_2Enum_2Enum > $o] :
( ( ? [V4l: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esums @ V1g @ V4l @ V3s )
& ? [V5N: tyop_2Enum_2Enum] :
! [V6n: tyop_2Enum_2Enum,V7x: A_27a] :
( ( ( c_2Earithmetic_2E_3C_3D @ V5N @ V6n )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V6n @ V3s )
& ( V2P @ V7x ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V7x @ V6n ) ) @ ( V1g @ V6n ) ) ) )
=> ? [V8l: A_27a > tyop_2Erealax_2Ereal] :
! [V9e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V9e )
=> ? [V10N: tyop_2Enum_2Enum] :
! [V11n: tyop_2Enum_2Enum,V12x: A_27a] :
( ( ( c_2Earithmetic_2E_3C_3D @ V10N @ V11n )
& ( V2P @ V12x ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V3s @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V11n ) ) @ ( V0f @ V12x ) ) @ ( V8l @ V12x ) ) ) @ V9e ) ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__COMPARISON__BOUND,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Esums @ V1g @ V3a @ V2s )
& ! [V4i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4i @ V2s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V4i ) ) @ ( V1g @ V4i ) ) ) )
=> ? [V5l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esums @ V0f @ V5l @ V2s )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V5l ) @ V3a ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__SUBSET__ABSCONV,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2t: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esummable @ V1s
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0x @ V3n ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Enum_2Enum @ V2t @ V1s ) )
=> ( c_2Ereal__topology_2Esummable @ V2t
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0x @ V4n ) ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__ABSCONV__IMP__CONV,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esummable @ V1k
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0x @ V2n ) ) )
=> ( c_2Ereal__topology_2Esummable @ V1k @ V0x ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__COMPARISON,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esummable @ V2s @ V1g )
& ? [V3N: tyop_2Enum_2Enum] :
! [V4n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3E_3D @ V4n @ V3N )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4n @ V2s ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V4n ) ) @ ( V1g @ V4n ) ) ) )
=> ( c_2Ereal__topology_2Esummable @ V2s @ V0f ) ) ).
thf(thm_2Ereal__topology_2ESERIES__COMPARISON,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o] :
( ( ? [V3l: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esums @ V1g @ V3l @ V2s )
& ? [V4N: tyop_2Enum_2Enum] :
! [V5n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3E_3D @ V5n @ V4N )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5n @ V2s ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V5n ) ) @ ( V1g @ V5n ) ) ) )
=> ? [V6l: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esums @ V0f @ V6l @ V2s ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__IMP__SUMS__BOUNDED,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Esummable @ ( c_2Ereal__topology_2Efrom @ V1k ) @ V0f )
=> ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1k @ V2n ) @ V0f ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__IMP__BOUNDED,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esummable @ V1k @ V0f )
=> ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ V0f @ V1k ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__IMP__TOZERO,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esummable @ V1k @ V0f )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2n @ V1k ) @ ( V0f @ V2n ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ c_2Ereal__topology_2Esequentially ) ) ).
thf(thm_2Ereal__topology_2ESERIES__GOESTOZERO,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esummable @ V0s @ V1x )
=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ( c_2Ereal__topology_2Eeventually @ tyop_2Enum_2Enum
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3n @ V0s ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( V1x @ V3n ) ) @ V2e ) )
@ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__CAUCHY__UNIFORM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2k: tyop_2Enum_2Enum > $o] :
( ? [V3l: A_27a > tyop_2Erealax_2Ereal] :
! [V4e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4e )
=> ? [V5N: tyop_2Enum_2Enum] :
! [V6n: tyop_2Enum_2Enum,V7x: A_27a] :
( ( ( c_2Earithmetic_2E_3C_3D @ V5N @ V6n )
& ( V0P @ V7x ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V2k @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V6n ) ) @ ( V1f @ V7x ) ) @ ( V3l @ V7x ) ) ) @ V4e ) ) )
<=> ! [V8e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V8e )
=> ? [V9N: tyop_2Enum_2Enum] :
! [V10m: tyop_2Enum_2Enum,V11n: tyop_2Enum_2Enum,V12x: A_27a] :
( ( ( c_2Earithmetic_2E_3C_3D @ V9N @ V10m )
& ( V0P @ V12x ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V2k @ ( c_2Eiterate_2E_2E_2E @ V10m @ V11n ) ) @ ( V1f @ V12x ) ) ) @ V8e ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__FROM__ELSEWHERE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Esummable @ ( c_2Ereal__topology_2Efrom @ V1m ) @ V0f )
=> ( c_2Ereal__topology_2Esummable @ ( c_2Ereal__topology_2Efrom @ V2n ) @ V0f ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__EQ__COFINITE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2t: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ V1s @ V2t ) @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ V2t @ V1s ) ) )
& ( c_2Ereal__topology_2Esummable @ V1s @ V0f ) )
=> ( c_2Ereal__topology_2Esummable @ V2t @ V0f ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__IFF__COFINITE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2t: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ V1s @ V2t ) @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ V2t @ V1s ) ) )
=> ( ( c_2Ereal__topology_2Esummable @ V1s @ V0f )
= ( c_2Ereal__topology_2Esummable @ V2t @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__EQ__EVENTUALLY,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2k: tyop_2Enum_2Enum > $o] :
( ( ? [V3N: tyop_2Enum_2Enum] :
! [V4n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V3N @ V4n )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4n @ V2k ) )
=> ( ( V0f @ V4n )
= ( V1g @ V4n ) ) )
& ( c_2Ereal__topology_2Esummable @ V2k @ V0f ) )
=> ( c_2Ereal__topology_2Esummable @ V2k @ V1g ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__IFF__EVENTUALLY,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2k: tyop_2Enum_2Enum > $o] :
( ? [V3N: tyop_2Enum_2Enum] :
! [V4n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V3N @ V4n )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4n @ V2k ) )
=> ( ( V0f @ V4n )
= ( V1g @ V4n ) ) )
=> ( ( c_2Ereal__topology_2Esummable @ V2k @ V0f )
= ( c_2Ereal__topology_2Esummable @ V2k @ V1g ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__CAUCHY,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esummable @ V1s @ V0f )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3N: tyop_2Enum_2Enum] :
! [V4m: tyop_2Enum_2Enum,V5n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V4m @ V3N )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V1s @ ( c_2Eiterate_2E_2E_2E @ V4m @ V5n ) ) @ V0f ) ) @ V2e ) ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__CAUCHY,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o] :
( ? [V2l: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esums @ V0f @ V2l @ V1s )
<=> ! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4N: tyop_2Enum_2Enum] :
! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V5m @ V4N )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V1s @ ( c_2Eiterate_2E_2E_2E @ V5m @ V6n ) ) @ V0f ) ) @ V3e ) ) ) ) ).
thf(thm_2Ereal__topology_2EABS__SUM__TRIVIAL__LEMMA,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1n: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V4P: $o,V5e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5e )
=> ( ( V4P
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V0s @ ( c_2Eiterate_2E_2E_2E @ V2m @ V1n ) ) @ V3f ) ) @ V5e ) )
<=> ( V4P
=> ( ( c_2Eprim__rec_2E_3C @ V1n @ V2m )
| ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V0s @ ( c_2Eiterate_2E_2E_2E @ V2m @ V1n ) ) @ V3f ) ) @ V5e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUM__DIFF__LEMMA,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k: tyop_2Enum_2Enum > $o,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V2m @ V3n )
=> ( ( c_2Ereal_2Ereal__sub @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V1k @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V3n ) ) @ V0f ) @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V1k @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V2m ) ) @ V0f ) )
= ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V1k @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V2m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V3n ) ) @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2ESEQUENCE__CAUCHY__WLOG,axiom,
! [V0e: tyop_2Erealax_2Ereal,V1P: tyop_2Enum_2Enum > $o,V2s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( ( V1P @ V3m )
& ( V1P @ V4n ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V2s @ V3m ) @ ( V2s @ V4n ) ) ) @ V0e ) )
<=> ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
( ( ( V1P @ V5m )
& ( V1P @ V6n )
& ( c_2Earithmetic_2E_3C_3D @ V5m @ V6n ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V2s @ V5m ) @ ( V2s @ V6n ) ) ) @ V0e ) ) ) ).
thf(thm_2Ereal__topology_2EPARTIAL__SUMS__DROP__LE__INFSUM,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2n: tyop_2Enum_2Enum] :
( ( ! [V3i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3i @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V3i ) ) )
& ( c_2Ereal__topology_2Esummable @ V1s @ V0f ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V1s @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V2n ) ) @ V0f ) @ ( c_2Ereal__topology_2Einfsum @ V1s @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2EPARTIAL__SUMS__COMPONENT__LE__INFSUM,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2n: tyop_2Enum_2Enum] :
( ( ! [V3i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3i @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V3i ) ) )
& ( c_2Ereal__topology_2Esummable @ V1s @ V0f ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V1s @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V2n ) ) @ V0f ) @ ( c_2Ereal__topology_2Einfsum @ V1s @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2EINFSUM__RESTRICT,axiom,
! [V0k: tyop_2Enum_2Enum > $o,V1a: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Einfsum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum )
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2n @ V0k ) @ ( V1a @ V2n ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
= ( c_2Ereal__topology_2Einfsum @ V0k @ V1a ) ) ).
thf(thm_2Ereal__topology_2EINFSUM__EQ,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2k: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esummable @ V2k @ V0f )
& ( c_2Ereal__topology_2Esummable @ V2k @ V1g )
& ! [V3x: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3x @ V2k )
=> ( ( V0f @ V3x )
= ( V1g @ V3x ) ) ) )
=> ( ( c_2Ereal__topology_2Einfsum @ V2k @ V0f )
= ( c_2Ereal__topology_2Einfsum @ V2k @ V1g ) ) ) ).
thf(thm_2Ereal__topology_2EINFSUM__NEG,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esummable @ V0s @ V1x )
=> ( ( c_2Ereal__topology_2Einfsum @ V0s
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( V1x @ V2n ) ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Ereal__topology_2Einfsum @ V0s @ V1x ) ) ) ) ).
thf(thm_2Ereal__topology_2EINFSUM__CMUL,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esummable @ V0s @ V1x )
=> ( ( c_2Ereal__topology_2Einfsum @ V0s
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ V2c @ ( V1x @ V3n ) ) )
= ( c_2Erealax_2Ereal__mul @ V2c @ ( c_2Ereal__topology_2Einfsum @ V0s @ V1x ) ) ) ) ).
thf(thm_2Ereal__topology_2EINFSUM__SUB,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esummable @ V2s @ V0x )
& ( c_2Ereal__topology_2Esummable @ V2s @ V1y ) )
=> ( ( c_2Ereal__topology_2Einfsum @ V2s
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0x @ V3i ) @ ( V1y @ V3i ) ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Ereal__topology_2Einfsum @ V2s @ V0x ) @ ( c_2Ereal__topology_2Einfsum @ V2s @ V1y ) ) ) ) ).
thf(thm_2Ereal__topology_2EINFSUM__ADD,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esummable @ V2s @ V0x )
& ( c_2Ereal__topology_2Esummable @ V2s @ V1y ) )
=> ( ( c_2Ereal__topology_2Einfsum @ V2s
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add @ ( V0x @ V3i ) @ ( V1y @ V3i ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2Einfsum @ V2s @ V0x ) @ ( c_2Ereal__topology_2Einfsum @ V2s @ V1y ) ) ) ) ).
thf(thm_2Ereal__topology_2EINFSUM__0,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Einfsum @ V0s
@ ^ [V1i: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal__topology_2EINFSUM__LINEAR,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esummable @ V2s @ V0f )
& ( c_2Ereal__topology_2Elinear @ V1h ) )
=> ( ( c_2Ereal__topology_2Einfsum @ V2s
@ ^ [V3n: tyop_2Enum_2Enum] : ( V1h @ ( V0f @ V3n ) ) )
= ( V1h @ ( c_2Ereal__topology_2Einfsum @ V2s @ V0f ) ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__BOUND,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Erealax_2Ereal,V4b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Esums @ V0f @ V3a @ V2s )
& ( c_2Ereal__topology_2Esums @ V1g @ V4b @ V2s )
& ! [V5i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5i @ V2s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V5i ) ) @ ( V1g @ V5i ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V3a ) @ V4b ) ) ).
thf(thm_2Ereal__topology_2ESERIES__DROP__POS,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Esums @ V0f @ V2a @ V1s )
& ! [V3x: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3x @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V3x ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2a ) ) ).
thf(thm_2Ereal__topology_2ESERIES__DROP__LE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Erealax_2Ereal,V4b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Esums @ V0f @ V3a @ V2s )
& ( c_2Ereal__topology_2Esums @ V1g @ V4b @ V2s )
& ! [V5x: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5x @ V2s )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V5x ) @ ( V1g @ V5x ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ V3a @ V4b ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__REINDEX,axiom,
! [V0k: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Esummable @ ( c_2Ereal__topology_2Efrom @ V2n )
@ ^ [V3x: tyop_2Enum_2Enum] : ( V1a @ ( c_2Earithmetic_2E_2B @ V3x @ V0k ) ) )
= ( c_2Ereal__topology_2Esummable @ ( c_2Ereal__topology_2Efrom @ ( c_2Earithmetic_2E_2B @ V2n @ V0k ) ) @ V1a ) ) ).
thf(thm_2Ereal__topology_2ESUMS__OFFSET__REV,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( ( c_2Ereal__topology_2Esums @ V0f @ V1l @ ( c_2Ereal__topology_2Efrom @ V2m ) )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2m )
& ( c_2Earithmetic_2E_3C_3D @ V3n @ V2m ) )
=> ( c_2Ereal__topology_2Esums @ V0f @ ( c_2Erealax_2Ereal__add @ V1l @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V3n @ ( c_2Earithmetic_2E_2D @ V2m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0f ) ) @ ( c_2Ereal__topology_2Efrom @ V3n ) ) ) ).
thf(thm_2Ereal__topology_2ESUMS__OFFSET,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( ( c_2Ereal__topology_2Esums @ V0f @ V1l @ ( c_2Ereal__topology_2Efrom @ V2m ) )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V3n )
& ( c_2Earithmetic_2E_3C_3D @ V2m @ V3n ) )
=> ( c_2Ereal__topology_2Esums @ V0f @ ( c_2Ereal_2Ereal__sub @ V1l @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ ( c_2Earithmetic_2E_2D @ V3n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0f ) ) @ ( c_2Ereal__topology_2Efrom @ V3n ) ) ) ).
thf(thm_2Ereal__topology_2ESUMS__FINITE__UNION,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2t: tyop_2Enum_2Enum > $o,V3l: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V2t )
& ( c_2Ereal__topology_2Esums @ V0f @ V3l @ V1s ) )
=> ( c_2Ereal__topology_2Esums @ V0f @ ( c_2Erealax_2Ereal__add @ V3l @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ V2t @ V1s ) @ V0f ) ) @ ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ V1s @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2ESUMS__FINITE__DIFF,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1t: tyop_2Enum_2Enum > $o,V2s: tyop_2Enum_2Enum > $o,V3l: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Enum_2Enum @ V1t @ V2s )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1t )
& ( c_2Ereal__topology_2Esums @ V0f @ V3l @ V2s ) )
=> ( c_2Ereal__topology_2Esums @ V0f @ ( c_2Ereal_2Ereal__sub @ V3l @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ V1t @ V0f ) ) @ ( c_2Epred__set_2EDIFF @ tyop_2Enum_2Enum @ V2s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__RESTRICT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esummable @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum )
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2n @ V1k ) @ ( V0f @ V2n ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
= ( c_2Ereal__topology_2Esummable @ V1k @ V0f ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__TRIVIAL,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esummable @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) @ V0f ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__SUBSET,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2t: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Enum_2Enum @ V1s @ V2t )
& ( c_2Ereal__topology_2Esummable @ V2t
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3i @ V1s ) @ ( V0x @ V3i ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Ereal__topology_2Esummable @ V1s @ V0x ) ) ).
thf(thm_2Ereal__topology_2ESERIES__SUBSET,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2t: tyop_2Enum_2Enum > $o,V3l: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Enum_2Enum @ V1s @ V2t )
& ( c_2Ereal__topology_2Esums
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4i @ V1s ) @ ( V0x @ V4i ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
@ V3l
@ V2t ) )
=> ( c_2Ereal__topology_2Esums @ V0x @ V3l @ V1s ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__COMPONENT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esummable @ V1s @ V0f )
=> ( c_2Ereal__topology_2Esummable @ V1s
@ ^ [V2i: tyop_2Enum_2Enum] : ( V0f @ V2i ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__EQ,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2k: tyop_2Enum_2Enum > $o] :
( ( ! [V3x: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3x @ V2k )
=> ( ( V0f @ V3x )
= ( V1g @ V3x ) ) )
& ( c_2Ereal__topology_2Esummable @ V2k @ V0f ) )
=> ( c_2Ereal__topology_2Esummable @ V2k @ V1g ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__IFF,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2k: tyop_2Enum_2Enum > $o] :
( ! [V3x: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3x @ V2k )
=> ( ( V0f @ V3x )
= ( V1g @ V3x ) ) )
=> ( ( c_2Ereal__topology_2Esummable @ V2k @ V0f )
= ( c_2Ereal__topology_2Esummable @ V2k @ V1g ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__NEG,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esummable @ V1s @ V0x )
=> ( c_2Ereal__topology_2Esummable @ V1s
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( V0x @ V2n ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__CMUL,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esummable @ V0s @ V1x )
=> ( c_2Ereal__topology_2Esummable @ V0s
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ V2c @ ( V1x @ V3n ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__SUB,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esummable @ V2s @ V0x )
& ( c_2Ereal__topology_2Esummable @ V2s @ V1y ) )
=> ( c_2Ereal__topology_2Esummable @ V2s
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0x @ V3n ) @ ( V1y @ V3n ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__ADD,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esummable @ V2s @ V0x )
& ( c_2Ereal__topology_2Esummable @ V2s @ V1y ) )
=> ( c_2Ereal__topology_2Esummable @ V2s
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add @ ( V0x @ V3n ) @ ( V1y @ V3n ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__0,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( c_2Ereal__topology_2Esummable @ V0s
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal__topology_2ESUMMABLE__LINEAR,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esummable @ V2s @ V0f )
& ( c_2Ereal__topology_2Elinear @ V1h ) )
=> ( c_2Ereal__topology_2Esummable @ V2s
@ ^ [V3n: tyop_2Enum_2Enum] : ( V1h @ ( V0f @ V3n ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUMS__REINDEX__GEN,axiom,
! [V0k: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3s: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esums
@ ^ [V4x: tyop_2Enum_2Enum] : ( V1a @ ( c_2Earithmetic_2E_2B @ V4x @ V0k ) )
@ V2l
@ V3s )
= ( c_2Ereal__topology_2Esums @ V1a @ V2l
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V5i: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ V5i @ V0k )
@ V3s ) ) ) ).
thf(thm_2Ereal__topology_2ESUMS__REINDEX,axiom,
! [V0k: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Esums
@ ^ [V4x: tyop_2Enum_2Enum] : ( V1a @ ( c_2Earithmetic_2E_2B @ V4x @ V0k ) )
@ V2l
@ ( c_2Ereal__topology_2Efrom @ V3n ) )
= ( c_2Ereal__topology_2Esums @ V1a @ V2l @ ( c_2Ereal__topology_2Efrom @ ( c_2Earithmetic_2E_2B @ V3n @ V0k ) ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__SUM,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2k: tyop_2Enum_2Enum > $o,V3s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V3s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Enum_2Enum @ V3s @ V2k )
& ! [V4x: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4x @ V3s ) )
=> ( ( V0f @ V4x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ V3s @ V0f )
= V1l ) )
=> ( c_2Ereal__topology_2Esums @ V0f @ V1l @ V2k ) ) ).
thf(thm_2Ereal__topology_2ESERIES__RESTRICT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k: tyop_2Enum_2Enum > $o,V2l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esums
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3n @ V1k ) @ ( V0f @ V3n ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
@ V2l
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) )
= ( c_2Ereal__topology_2Esums @ V0f @ V2l @ V1k ) ) ).
thf(thm_2Ereal__topology_2ESERIES__TRIVIAL,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esums @ V0f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ).
thf(thm_2Ereal__topology_2ESERIES__DIFFS,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ c_2Ereal__topology_2Esequentially )
=> ( c_2Ereal__topology_2Esums
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0f @ V2n ) @ ( V0f @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
@ ( V0f @ V1k )
@ ( c_2Ereal__topology_2Efrom @ V1k ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__COMPONENT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esums @ V0f @ V2l @ V1s )
=> ( c_2Ereal__topology_2Esums
@ ^ [V3i: tyop_2Enum_2Enum] : ( V0f @ V3i )
@ V2l
@ V1s ) ) ).
thf(thm_2Ereal__topology_2ESERIES__FINITE__SUPPORT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2k: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V1s @ V2k ) )
& ! [V3x: tyop_2Enum_2Enum] :
( ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3x @ V2k )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3x @ V1s ) ) )
=> ( ( V0f @ V3x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Ereal__topology_2Esums @ V0f @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V1s @ V2k ) @ V0f ) @ V2k ) ) ).
thf(thm_2Ereal__topology_2ESUMS__0,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o] :
( ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2n @ V1s )
=> ( ( V0f @ V2n )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( c_2Ereal__topology_2Esums @ V0f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1s ) ) ).
thf(thm_2Ereal__topology_2ESUMS__EQ,axiom,
! [V0l: tyop_2Erealax_2Ereal,V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3k: tyop_2Enum_2Enum > $o] :
( ( ! [V4x: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4x @ V3k )
=> ( ( V1f @ V4x )
= ( V2g @ V4x ) ) )
& ( c_2Ereal__topology_2Esums @ V1f @ V0l @ V3k ) )
=> ( c_2Ereal__topology_2Esums @ V2g @ V0l @ V3k ) ) ).
thf(thm_2Ereal__topology_2ESUMS__IFF,axiom,
! [V0l: tyop_2Erealax_2Ereal,V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3k: tyop_2Enum_2Enum > $o] :
( ! [V4x: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4x @ V3k )
=> ( ( V1f @ V4x )
= ( V2g @ V4x ) ) )
=> ( ( c_2Ereal__topology_2Esums @ V1f @ V0l @ V3k )
= ( c_2Ereal__topology_2Esums @ V2g @ V0l @ V3k ) ) ) ).
thf(thm_2Ereal__topology_2ESERIES__NEG,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esums @ V0x @ V1x0 @ V2s )
=> ( c_2Ereal__topology_2Esums
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( V0x @ V3n ) )
@ ( c_2Erealax_2Ereal__neg @ V1x0 )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2ESERIES__CMUL,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3s: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esums @ V0x @ V1x0 @ V3s )
=> ( c_2Ereal__topology_2Esums
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ V2c @ ( V0x @ V4n ) )
@ ( c_2Erealax_2Ereal__mul @ V2c @ V1x0 )
@ V3s ) ) ).
thf(thm_2Ereal__topology_2ESERIES__SUB,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3y0: tyop_2Erealax_2Ereal,V4s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esums @ V0x @ V1x0 @ V4s )
& ( c_2Ereal__topology_2Esums @ V2y @ V3y0 @ V4s ) )
=> ( c_2Ereal__topology_2Esums
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0x @ V5n ) @ ( V2y @ V5n ) )
@ ( c_2Ereal_2Ereal__sub @ V1x0 @ V3y0 )
@ V4s ) ) ).
thf(thm_2Ereal__topology_2ESERIES__ADD,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3y0: tyop_2Erealax_2Ereal,V4s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esums @ V0x @ V1x0 @ V4s )
& ( c_2Ereal__topology_2Esums @ V2y @ V3y0 @ V4s ) )
=> ( c_2Ereal__topology_2Esums
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add @ ( V0x @ V5n ) @ ( V2y @ V5n ) )
@ ( c_2Erealax_2Ereal__add @ V1x0 @ V3y0 )
@ V4s ) ) ).
thf(thm_2Ereal__topology_2ESERIES__0,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( c_2Ereal__topology_2Esums
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0s ) ).
thf(thm_2Ereal__topology_2ESERIES__LINEAR,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esums @ V0f @ V2l @ V3s )
& ( c_2Ereal__topology_2Elinear @ V1h ) )
=> ( c_2Ereal__topology_2Esums
@ ^ [V4n: tyop_2Enum_2Enum] : ( V1h @ ( V0f @ V4n ) )
@ ( V1h @ V2l )
@ V3s ) ) ).
thf(thm_2Ereal__topology_2ESERIES__FINITE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s )
=> ( c_2Ereal__topology_2Esums @ V0f @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ V1s @ V0f ) @ V1s ) ) ).
thf(thm_2Ereal__topology_2ESERIES__TERMS__TOZERO,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Esums @ V0f @ V1l @ ( c_2Ereal__topology_2Efrom @ V2n ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ c_2Ereal__topology_2Esequentially ) ) ).
thf(thm_2Ereal__topology_2EINFSUM__UNIQUE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esums @ V0f @ V1l @ V2s )
=> ( ( c_2Ereal__topology_2Einfsum @ V2s @ V0f )
= V1l ) ) ).
thf(thm_2Ereal__topology_2ESERIES__UNIQUE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2l_27: tyop_2Erealax_2Ereal,V3s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ereal__topology_2Esums @ V0f @ V1l @ V3s )
& ( c_2Ereal__topology_2Esums @ V0f @ V2l_27 @ V3s ) )
=> ( V1l = V2l_27 ) ) ).
thf(thm_2Ereal__topology_2ESERIES__FROM,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2k: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Esums @ V0f @ V1l @ ( c_2Ereal__topology_2Efrom @ V2k ) )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2k @ V3n ) @ V0f )
@ V1l
@ c_2Ereal__topology_2Esequentially ) ) ).
thf(thm_2Ereal__topology_2EFINITE__INTER__NUMSEG,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] : ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V0s @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) ) ) ).
thf(thm_2Ereal__topology_2ESUMS__LIM,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esums @ V0f
@ ( c_2Ereal__topology_2Elim__def @ tyop_2Enum_2Enum @ c_2Ereal__topology_2Esequentially
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V1s @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V2n ) ) @ V0f ) )
@ V1s )
= ( c_2Ereal__topology_2Esummable @ V1s @ V0f ) ) ).
thf(thm_2Ereal__topology_2ESUMS__INFSUM,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esums @ V0f @ ( c_2Ereal__topology_2Einfsum @ V1s @ V0f ) @ V1s )
= ( c_2Ereal__topology_2Esummable @ V1s @ V0f ) ) ).
thf(thm_2Ereal__topology_2ESUMS__SUMMABLE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2s: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Esums @ V0f @ V1l @ V2s )
=> ( c_2Ereal__topology_2Esummable @ V2s @ V0f ) ) ).
thf(thm_2Ereal__topology_2EIMAGE__AFFINITY__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2m: tyop_2Erealax_2Ereal,V3c: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V2m @ V4x ) @ V3c )
@ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) @ ( c_2Ebool_2ECOND @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2m ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V2m @ V0a ) @ V3c ) @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V2m @ V1b ) @ V3c ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V2m @ V1b ) @ V3c ) @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V2m @ V0a ) @ V3c ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__EQ__AFFINITY,axiom,
! [V0m: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0m
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( V3y
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0m @ V2x ) @ V1c ) )
<=> ( ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0m ) @ V3y ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ V1c @ V0m ) ) )
= V2x ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__AFFINITY__EQ,axiom,
! [V0m: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0m
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0m @ V2x ) @ V1c )
= V3y )
<=> ( V2x
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0m ) @ V3y ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ V1c @ V0m ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LT__AFFINITY,axiom,
! [V0m: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0m )
=> ( ( c_2Erealax_2Ereal__lt @ V3y @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0m @ V2x ) @ V1c ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0m ) @ V3y ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ V1c @ V0m ) ) ) @ V2x ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__AFFINITY__LT,axiom,
! [V0m: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0m )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0m @ V2x ) @ V1c ) @ V3y )
= ( c_2Erealax_2Ereal__lt @ V2x @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0m ) @ V3y ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ V1c @ V0m ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LE__AFFINITY,axiom,
! [V0m: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0m )
=> ( ( c_2Ereal_2Ereal__lte @ V3y @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0m @ V2x ) @ V1c ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0m ) @ V3y ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ V1c @ V0m ) ) ) @ V2x ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__AFFINITY__LE,axiom,
! [V0m: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0m )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0m @ V2x ) @ V1c ) @ V3y )
= ( c_2Ereal_2Ereal__lte @ V2x @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0m ) @ V3y ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ V1c @ V0m ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EAFFINITY__INVERSES,axiom,
! [V0m: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0m
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0m @ V2x ) @ V1c )
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0m ) @ V3x ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0m ) @ V1c ) ) ) )
= ( ^ [V4x: tyop_2Erealax_2Ereal] : V4x ) )
& ( ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0m ) @ V5x ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0m ) @ V1c ) ) )
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0m @ V6x ) @ V1c ) )
= ( ^ [V7x: tyop_2Erealax_2Ereal] : V7x ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIM__SUBSTANDARD,axiom,
( ( c_2Ereal__topology_2Edim
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V0x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) )
= c_2Enum_2E0 ) ).
thf(thm_2Ereal__topology_2ECLOSED__SUBSTANDARD,axiom,
( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V0x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__SUBSTANDARD,axiom,
( c_2Ereal__topology_2Esubspace
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V0x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESYMMETRIC__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V1x ) @ V0s ) )
=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V2x ) @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__NEGATIONS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eclosure
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V1x )
@ V0s ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V2x )
@ ( c_2Ereal__topology_2Eclosure @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ESYMMETRIC__INTERIOR,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V1x ) @ V0s ) )
=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Einterior @ V0s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V2x ) @ ( c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__NEGATIONS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Einterior
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V1x )
@ V0s ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V2x )
@ ( c_2Ereal__topology_2Einterior @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__INJECTIVE__LINEAR__IMAGE__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V1f )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( V1f @ V3x )
= ( V1f @ V4y ) )
=> ( V3x = V4y ) ) )
=> ( ( c_2Ereal__topology_2Elimit__point__of @ ( V1f @ V0x ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1f @ V2s ) )
= ( c_2Ereal__topology_2Elimit__point__of @ V0x @ V2s ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPLETE__INJECTIVE__LINEAR__IMAGE__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V2x )
= ( V0f @ V3y ) )
=> ( V2x = V3y ) ) )
=> ( ( c_2Ereal__topology_2Ecomplete @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
= ( c_2Ereal__topology_2Ecomplete @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPLETE__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V1x )
= ( V0f @ V2y ) )
=> ( V1x = V2y ) ) )
=> ! [V3s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecomplete @ V3s )
=> ( c_2Ereal__topology_2Ecomplete @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3s ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V2x )
= ( V0f @ V3y ) )
=> ( V2x = V3y ) ) )
=> ( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Einterior @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__OPEN__MAPPING,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Elinear @ V1g )
& ( ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1g )
= ( c_2Ecombin_2EI @ tyop_2Erealax_2Ereal ) ) )
=> ! [V2s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V2s )
=> ( c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) ) ) ) ).
thf(thm_2Ereal__topology_2EIN__INTERIOR__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Elinear @ V1g )
& ( ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1g )
= ( c_2Ecombin_2EI @ tyop_2Erealax_2Ereal ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Ereal__topology_2Einterior @ V2s ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__BIJECTIVE__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V2x )
= ( V0f @ V3y ) )
=> ( V2x = V3y ) )
& ! [V4y: tyop_2Erealax_2Ereal] :
? [V5x: tyop_2Erealax_2Ereal] :
( ( V0f @ V5x )
= V4y ) )
=> ( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Einterior @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__IMAGE__SUBSET__INTERIOR,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2y: tyop_2Erealax_2Ereal] :
? [V3x: tyop_2Erealax_2Ereal] :
( ( V0f @ V3x )
= V2y ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Einterior @ V1s ) ) @ ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__INTERIOR__IMAGE__SUBSET,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V2x )
= ( V0f @ V3y ) )
=> ( V2x = V3y ) ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Einterior @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__BOUNDED__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Ebounded__def @ V1s ) )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V2x )
= ( V0f @ V3y ) )
=> ( V2x = V3y ) ) )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__LINEAR__IMAGE__SUBSET,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__INJECTIVE__LINEAR__IMAGE__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V2x )
= ( V0f @ V3y ) )
=> ( V2x = V3y ) ) )
=> ( ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
= ( c_2Ereal__topology_2EClosed @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V1x )
= ( V0f @ V2y ) )
=> ( V1x = V2y ) ) )
=> ! [V3s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V3s )
=> ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3s ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__BIJECTIVE__LINEAR__IMAGE__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V2x )
= ( V0f @ V3y ) )
=> ( V2x = V3y ) )
& ! [V4y: tyop_2Erealax_2Ereal] :
? [V5x: tyop_2Erealax_2Ereal] :
( ( V0f @ V5x )
= V4y ) )
=> ( ( c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
= ( c_2Ereal__topology_2EOpen @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__SURJECTIVE__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V1y: tyop_2Erealax_2Ereal] :
? [V2x: tyop_2Erealax_2Ereal] :
( ( V0f @ V2x )
= V1y ) )
=> ! [V3s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V3s )
=> ( c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__INJECTIVE__IMAGE__SUBSPACE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Esubspace @ V1s )
& ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
& ( ( V0f @ V2x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( V2x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( c_2Ereal__topology_2EClosed @ V1s ) )
=> ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EINJECTIVE__IMP__ISOMETRIC,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V1s )
& ( c_2Ereal__topology_2Esubspace @ V1s )
& ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
& ( ( V0f @ V2x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( V2x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ? [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s )
=> ( c_2Ereal_2Ereal__ge @ ( c_2Ereal_2Eabs @ ( V0f @ V4x ) ) @ ( c_2Erealax_2Ereal__mul @ V3e @ ( c_2Ereal_2Eabs @ V4x ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPLETE__ISOMETRIC__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ( c_2Ereal__topology_2Esubspace @ V1s )
& ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
=> ( c_2Ereal_2Ereal__ge @ ( c_2Ereal_2Eabs @ ( V0f @ V3x ) ) @ ( c_2Erealax_2Ereal__mul @ V2e @ ( c_2Ereal_2Eabs @ V3x ) ) ) )
& ( c_2Ereal__topology_2Ecomplete @ V1s ) )
=> ( c_2Ereal__topology_2Ecomplete @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECAUCHY__ISOMETRIC,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2e: tyop_2Erealax_2Ereal,V3x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ( c_2Ereal__topology_2Esubspace @ V1s )
& ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s )
=> ( c_2Ereal_2Ereal__ge @ ( c_2Ereal_2Eabs @ ( V0f @ V4x ) ) @ ( c_2Erealax_2Ereal__mul @ V2e @ ( c_2Ereal_2Eabs @ V4x ) ) ) )
& ! [V5n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V3x @ V5n ) @ V1s )
& ( c_2Ereal__topology_2Ecauchy @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3x ) ) )
=> ( c_2Ereal__topology_2Ecauchy @ V3x ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__STANDARD__HYPERPLANE__HYPERPLANE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0a
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( c_2Ereal__topology_2Ehomeomorphic
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V3x @ V2c ) ) )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0a @ V4x ) @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__HYPERPLANE__STANDARD__HYPERPLANE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0a
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( c_2Ereal__topology_2Ehomeomorphic
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0a @ V3x ) @ V1b ) ) )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V4x @ V2c ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__HYPERPLANES,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0a
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V2c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Ereal__topology_2Ehomeomorphic
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0a @ V4x ) @ V1b ) ) )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V2c @ V5x ) @ V3d ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__FINITE__RANGE__CONSTANT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V1s )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) )
=> ? [V2a: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
=> ( ( V0f @ V3x )
= V2a ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__DISCRETE__RANGE__CONSTANT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V1s )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
=> ? [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
& ! [V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1s )
& ( (~)
@ ( ( V0f @ V4y )
= ( V0f @ V2x ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ V3e @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V0f @ V4y ) @ ( V0f @ V2x ) ) ) ) ) ) ) )
=> ? [V5a: tyop_2Erealax_2Ereal] :
! [V6x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s )
=> ( ( V0f @ V6x )
= V5a ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__RANGE__CONSTANT,axiom,
! [V0t: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V2s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V2s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1f @ V2s ) @ V0t )
& ! [V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V0t )
=> ( ( c_2Ereal__topology_2Econnected__component @ V0t @ V3y )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ? [V4a: tyop_2Erealax_2Ereal] :
! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V2s )
=> ( ( V1f @ V5x )
= V4a ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__FINITE__RANGE__CONSTANT__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ! [V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V0s )
& ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1f @ V0s ) ) )
=> ? [V2a: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( ( V1f @ V3x )
= V2a ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__DISCRETE__RANGE__CONSTANT__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ! [V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V0s )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ? [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
& ! [V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V0s )
& ( (~)
@ ( ( V1f @ V4y )
= ( V1f @ V2x ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ V3e @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V1f @ V4y ) @ ( V1f @ V2x ) ) ) ) ) ) ) )
=> ? [V5a: tyop_2Erealax_2Ereal] :
! [V6x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V0s )
=> ( ( V1f @ V6x )
= V5a ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__RANGE__CONSTANT__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ! [V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1f @ V0s ) @ V2t )
& ! [V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V2t )
=> ( ( c_2Ereal__topology_2Econnected__component @ V2t @ V3y )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ? [V4a: tyop_2Erealax_2Ereal] :
! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V0s )
=> ( ( V1f @ V5x )
= V4a ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__DISCRETE__FINITE__RANGE__CONSTANT__EQ,axiom,
( ! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ! [V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1f @ V0s ) @ V2t )
& ! [V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V2t )
=> ( ( c_2Ereal__topology_2Econnected__component @ V2t @ V3y )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ? [V4a: tyop_2Erealax_2Ereal] :
! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V0s )
=> ( ( V1f @ V5x )
= V4a ) ) ) )
& ! [V6s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V6s )
<=> ! [V7f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V7f @ V6s )
& ! [V8x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V6s )
=> ? [V9e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V9e )
& ! [V10y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10y @ V6s )
& ( (~)
@ ( ( V7f @ V10y )
= ( V7f @ V8x ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ V9e @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V7f @ V10y ) @ ( V7f @ V8x ) ) ) ) ) ) ) )
=> ? [V11a: tyop_2Erealax_2Ereal] :
! [V12x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V12x @ V6s )
=> ( ( V7f @ V12x )
= V11a ) ) ) )
& ! [V13s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V13s )
<=> ! [V14f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V14f @ V13s )
& ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V14f @ V13s ) ) )
=> ? [V15a: tyop_2Erealax_2Ereal] :
! [V16x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V16x @ V13s )
=> ( ( V14f @ V16x )
= V15a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EFINITE__BALL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) )
= ( c_2Ereal_2Ereal__lte @ V1r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2EFINITE__CBALL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) )
= ( c_2Ereal_2Ereal__lte @ V1r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2EEMPTY__INTERIOR__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
=> ( ( c_2Ereal__topology_2Einterior @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IMP__INFINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EFINITE__IMP__NOT__OPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( (~) @ ( c_2Ereal__topology_2EOpen @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECARD__EQ__CBALL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1r )
=> ( c_2Ecardinal_2Ecardeq @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECARD__EQ__BALL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1r )
=> ( c_2Ecardinal_2Ecardeq @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EUNCOUNTABLE__OPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( (~) @ ( c_2Epred__set_2Ecountable @ tyop_2Erealax_2Ereal @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECARD__EQ__OPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ecardinal_2Ecardeq @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECOUNTABLE__OPEN__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2Ecountable @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
<=> ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EUNCOUNTABLE__INTERVAL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ( (~) @ ( c_2Epred__set_2Ecountable @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ( (~) @ ( c_2Epred__set_2Ecountable @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECARD__EQ__INTERVAL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ( c_2Ecardinal_2Ecardeq @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ( c_2Ecardinal_2Ecardeq @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EUNCOUNTABLE__EUCLIDEAN,axiom,
(~) @ ( c_2Epred__set_2Ecountable @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ECARD__EQ__EUCLIDEAN,axiom,
c_2Ecardinal_2Ecardeq @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2ECARD__EQ__REAL__IMP__UNCOUNTABLE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ecardinal_2Ecardeq @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
=> ( (~) @ ( c_2Epred__set_2Ecountable @ tyop_2Erealax_2Ereal @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EUNCOUNTABLE__REAL,axiom,
(~) @ ( c_2Epred__set_2Ecountable @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ECARD__EQ__REAL,axiom,
c_2Ecardinal_2Ecardeq @ tyop_2Erealax_2Ereal @ ( tyop_2Enum_2Enum > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Enum_2Enum > $o ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__OPEN__INTERVAL__UNIV,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0a @ V1b )
=> ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__OPEN__INTERVALS,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V0a @ V1b )
& ( c_2Erealax_2Ereal__lt @ V2c @ V3d ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__ONE__POINT__COMPACTIFICATIONS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2a @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3b @ V1t )
& ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Epred__set_2EDELETE @ tyop_2Erealax_2Ereal @ V0s @ V2a ) @ ( c_2Epred__set_2EDELETE @ tyop_2Erealax_2Ereal @ V1t @ V3b ) ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V1t ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__SPHERE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2d: tyop_2Erealax_2Ereal,V3e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2d )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2d ) ) @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V3e ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__CBALL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2d: tyop_2Erealax_2Ereal,V3e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2d )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2d ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V3e ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__BALLS,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2d: tyop_2Erealax_2Ereal,V3e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2d )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2d ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V3e ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__BALLS__CBALL__SPHERE,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2d: tyop_2Erealax_2Ereal,V3e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2d )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2d ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V3e ) ) ) )
& ! [V4a: tyop_2Erealax_2Ereal,V5b: tyop_2Erealax_2Ereal,V6d: tyop_2Erealax_2Ereal,V7e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V6d )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V7e ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V6d ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5b @ V7e ) ) ) )
& ! [V8a: tyop_2Erealax_2Ereal,V9b: tyop_2Erealax_2Ereal,V10d: tyop_2Erealax_2Ereal,V11e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V10d )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V11e ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V10d ) ) @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V9b @ V11e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__AFFINITY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V2c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ V0s
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V1a @ ( c_2Erealax_2Ereal__mul @ V2c @ V3x ) )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2Ehomeomorphic @ V0s
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V1a @ V2x )
@ V0s ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V1c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ V0s
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V1c @ V2x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__CONNECTEDNESS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V1t )
=> ( ( c_2Ereal__topology_2Econnected @ V0s )
= ( c_2Ereal__topology_2Econnected @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__COMPACTNESS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V1t )
=> ( ( c_2Ereal__topology_2Ecompact @ V0s )
= ( c_2Ereal__topology_2Ecompact @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__FROM__COMPOSITION__INJECTIVE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) @ V3t )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3t ) @ V4u )
& ! [V5x: tyop_2Erealax_2Ereal,V6y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V3t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6y @ V3t )
& ( ( V1g @ V5x )
= ( V1g @ V6y ) ) )
=> ( V5x = V6y ) )
& ? [V7h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V4u ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f ) @ V7h ) ) )
=> ( ? [V8f_27: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V3t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V8f_27 ) )
& ? [V9g_27: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V3t @ V4u ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V1g @ V9g_27 ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__FROM__COMPOSITION__SURJECTIVE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s )
= V3t )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3t ) @ V4u )
& ? [V5h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V4u ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f ) @ V5h ) ) )
=> ( ? [V6f_27: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V3t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V6f_27 ) )
& ? [V7g_27: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V3t @ V4u ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V1g @ V7g_27 ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V0s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1f @ V0s )
= V2t )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V0s )
& ( ( V1f @ V3x )
= ( V1f @ V4y ) ) )
=> ( V3x = V4y ) ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V2t ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V0s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1f @ V0s )
= V2t )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V0s )
& ( ( V1f @ V3x )
= ( V1f @ V4y ) ) )
=> ( V3x = V4y ) ) )
=> ? [V5g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V2t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V1f @ V5g ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Ecompact @ V2s )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V2s )
=> ( ( V1g @ ( V0f @ V3x ) )
= V3x ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V1g @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__IMP__QUOTIENT__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ( c_2Ereal__topology_2Ecompact @ V1s ) )
=> ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3u @ V2t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3u ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3u ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__IMP__CLOSED__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ( c_2Ereal__topology_2Ecompact @ V1s ) )
=> ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3u ) ) ) ) ).
thf(thm_2Ereal__topology_2EINJECTIVE__MAP__OPEN__IFF__CLOSED,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1s )
& ( ( V0f @ V3x )
= ( V0f @ V4y ) ) )
=> ( V3x = V4y ) ) )
=> ( ! [V5u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V5u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5u ) ) )
<=> ! [V6u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V6u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V6u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__CLOSED__MAP__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1s )
& ( ( V0f @ V3x )
= ( V0f @ V4y ) ) )
=> ( V3x = V4y ) ) )
=> ( ? [V5g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V2t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V5g ) )
<=> ! [V6u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V6u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V6u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__OPEN__MAP__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1s )
& ( ( V0f @ V3x )
= ( V0f @ V4y ) ) )
=> ( V3x = V4y ) ) )
=> ( ? [V5g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V2t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V5g ) )
<=> ! [V6u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V6u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V6u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__CLOSED__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V3t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V1g ) )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V4u ) )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V4u ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__OPEN__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V3t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V1g ) )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V4u ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V4u ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__CLOSED__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1s )
& ( ( V0f @ V3x )
= ( V0f @ V4y ) ) )
=> ( V3x = V4y ) )
& ! [V5u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V5u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5u ) ) ) )
=> ? [V6g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V2t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V6g ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__OPEN__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1s )
& ( ( V0f @ V3x )
= ( V0f @ V4y ) ) )
=> ( V3x = V4y ) )
& ! [V5u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V5u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5u ) ) ) )
=> ? [V6g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V2t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V6g ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE__CLOSED__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s )
= V3t )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V2s )
=> ( ( V1g @ ( V0f @ V4x ) )
= V4x ) )
& ! [V5u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V5u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5u ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3t ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE__OPEN__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s )
= V3t )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V2s )
=> ( ( V1g @ ( V0f @ V4x ) )
= V4x ) )
& ! [V5u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V5u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5u ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3t ) ) ).
thf(thm_2Ereal__topology_2ELIFT__TO__QUOTIENT__SPACE__UNIQUE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s )
= V3t )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V2s )
= V4u )
& ! [V5v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5v @ V3t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5v ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ V5v ) ) )
& ! [V7v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V7v @ V4u )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1g @ V8x ) @ V7v ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V4u ) @ V7v ) ) )
& ! [V9x: tyop_2Erealax_2Ereal,V10y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9x @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10y @ V2s ) )
=> ( ( ( V0f @ V9x )
= ( V0f @ V10y ) )
<=> ( ( V1g @ V9x )
= ( V1g @ V10y ) ) ) ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ V3t @ V4u ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__FINITE__STRONG,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
| ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1t ) )
=> ( ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V1t )
<=> ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1t )
& ( ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V0s )
= ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V1t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1t ) )
=> ( ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V1t )
<=> ( ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V0s )
= ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING__RIGHT,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0c )
=> ! [V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V1s
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V0c @ V3x )
@ V2t ) )
= ( c_2Ereal__topology_2Ehomeomorphic @ V1s @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING__LEFT,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0c )
=> ! [V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V0c @ V3x )
@ V1s )
@ V2t )
= ( c_2Ereal__topology_2Ehomeomorphic @ V1s @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__QUOTIENT__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V3t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V1g ) )
=> ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4u @ V3t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V4u ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ V4u ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__RIGHT__EQ,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V1s
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V3x )
@ V2t ) )
= ( c_2Ereal__topology_2Ehomeomorphic @ V1s @ V2t ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__LEFT__EQ,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V3x )
@ V1s )
@ V2t )
= ( c_2Ereal__topology_2Ehomeomorphic @ V1s @ V2t ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__SELF,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( c_2Ereal__topology_2Ehomeomorphic
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V2x )
@ V1s )
@ V1s ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__RIGHT__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V3x )
= ( V0f @ V4y ) )
=> ( V3x = V4y ) ) )
=> ( ( c_2Ereal__topology_2Ehomeomorphic @ V1s @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2t ) )
= ( c_2Ereal__topology_2Ehomeomorphic @ V1s @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__LEFT__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V3x )
= ( V0f @ V4y ) )
=> ( V3x = V4y ) ) )
=> ( ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
= ( c_2Ereal__topology_2Ehomeomorphic @ V1s @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__SELF,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V2x )
= ( V0f @ V3y ) )
=> ( V2x = V3y ) ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V1s ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__MINIMAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V1t )
<=> ? [V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V2f @ V4x ) @ V1t )
& ( ( V3g @ ( V2f @ V4x ) )
= V4x ) ) )
& ! [V5y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V1t )
=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V3g @ V5y ) @ V0s )
& ( ( V2f @ ( V3g @ V5y ) )
= V5y ) ) )
& ( c_2Ereal__topology_2Econtinuous__on @ V2f @ V0s )
& ( c_2Ereal__topology_2Econtinuous__on @ V3g @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__EMPTY,axiom,
( ! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) @ V1s )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__FINITENESS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V1t )
=> ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
= ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__IMP__CARD__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V1t )
=> ( c_2Ecardinal_2Ecardeq @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__TRANS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V1t )
& ( c_2Ereal__topology_2Ehomeomorphic @ V1t @ V2u ) )
=> ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V2u ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3k: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V4s: tyop_2Erealax_2Ereal > $o,V5t: tyop_2Erealax_2Ereal > $o,V6u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V4s @ V5t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V1g ) )
& ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V5t @ V6u ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V2h @ V3k ) ) )
=> ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V4s @ V6u ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2h @ V0f ) @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3k ) ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__SYM,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V1t )
= ( c_2Ereal__topology_2Ehomeomorphic @ V1t @ V0s ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__SYM,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V3t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V1g ) )
= ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V3t @ V2s ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V1g @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__REFL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V0s ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__ID,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V0s )
@ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal )
@ ^ [V1x: tyop_2Erealax_2Ereal] : V1x
@ ^ [V2x: tyop_2Erealax_2Ereal] : V2x ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM__OF__SUBSETS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4s_27: tyop_2Erealax_2Ereal > $o,V5t_27: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V3t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V1g ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4s_27 @ V2s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5t_27 @ V3t )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V4s_27 )
= V5t_27 ) )
=> ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V4s_27 @ V5t_27 ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V0f @ V1g ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHISM,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ehomeomorphism @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ V1t ) @ ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V2f @ V3g ) )
<=> ( ( c_2Ereal__topology_2Econtinuous__on @ V2f @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2f @ V0s ) @ V1t )
& ( c_2Ereal__topology_2Econtinuous__on @ V3g @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3g @ V1t ) @ V0s )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
=> ( ( V3g @ ( V2f @ V4x ) )
= V4x ) )
& ! [V5y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V1t )
=> ( ( V2f @ ( V3g @ V5y ) )
= V5y ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__DECREASING__CONVERGENT,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( V0s @ V1n ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
& ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0s @ ( c_2Enum_2ESUC @ V2n ) ) @ ( V0s @ V2n ) ) )
=> ? [V3l: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0s @ V3l @ c_2Ereal__topology_2Esequentially ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__INCREASING__CONVERGENT,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( V0s @ V1n ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
& ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0s @ V2n ) @ ( V0s @ ( c_2Enum_2ESUC @ V2n ) ) ) )
=> ? [V3l: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0s @ V3l @ c_2Ereal__topology_2Esequentially ) ) ).
thf(thm_2Ereal__topology_2EMAPPING__CONNECTED__ONTO__SEGMENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( (~)
@ ? [V3a: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ? [V4f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V4f @ V0s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4f @ V0s )
= ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__IVT__COMPONENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s )
& ( c_2Ereal_2Ereal__lte @ V1x @ V3a )
& ( c_2Ereal_2Ereal__lte @ V3a @ V2y ) )
=> ? [V4z: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4z @ V0s )
& ( V4z = V3a ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__IVT__HYPERPLANE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3a: tyop_2Erealax_2Ereal,V4b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s )
& ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V3a @ V1x ) @ V4b )
& ( c_2Ereal_2Ereal__lte @ V4b @ ( c_2Erealax_2Ereal__mul @ V3a @ V2y ) ) )
=> ? [V5z: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5z @ V0s )
& ( ( c_2Erealax_2Ereal__mul @ V3a @ V5z )
= V4b ) ) ) ).
thf(thm_2Ereal__topology_2EEXTENSION__FROM__CLOPEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V2t )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V2t )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2t ) @ V3u )
& ( ( V3u
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
=> ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V4g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V4g @ V1s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4g @ V1s ) @ V3u )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V2t )
=> ( ( V4g @ V5x )
= ( V0f @ V5x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__1,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4t: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4t @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4t @ V2s ) @ ( c_2Ereal_2Ereal__lte @ V4t @ V3a ) ) ) ) )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5t: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5t @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5t @ V2s ) @ ( c_2Ereal_2Ereal__lte @ V3a @ V5t ) ) ) ) )
& ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3a @ V2s )
=> ( ( V0f @ V3a )
= ( V1g @ V3a ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V6t: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__lte @ V6t @ V3a ) @ ( V0f @ V6t ) @ ( V1g @ V6t ) )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o,V4a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5t: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5t @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5t @ V3s ) @ ( c_2Ereal_2Ereal__lte @ ( V2h @ V5t ) @ V4a ) ) ) ) )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6t: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6t @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6t @ V3s ) @ ( c_2Ereal_2Ereal__lte @ V4a @ ( V2h @ V6t ) ) ) ) ) )
& ( c_2Ereal__topology_2Econtinuous__on @ V2h @ V3s )
& ! [V7t: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7t @ V3s )
& ( ( V2h @ V7t )
= V4a ) )
=> ( ( V0f @ V7t )
= ( V1g @ V7t ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V8t: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__lte @ ( V2h @ V8t ) @ V4a ) @ ( V0f @ V8t ) @ ( V1g @ V8t ) )
@ V3s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LOCAL,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o,V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V3s @ V4t ) ) @ V3s )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V3s @ V4t ) ) @ V4t )
& ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V3s )
& ( c_2Ereal__topology_2Econtinuous__on @ V2g @ V4t )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V3s )
& ( (~) @ ( V0P @ V5x ) ) )
| ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V4t )
& ( V0P @ V5x ) ) )
=> ( ( V1f @ V5x )
= ( V2g @ V5x ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( V0P @ V6x ) @ ( V1f @ V6x ) @ ( V2g @ V6x ) )
@ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V3s @ V4t ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__LOCAL,axiom,
! [V0t: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V2s @ V0t ) ) @ V2s )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V2s @ V0t ) ) @ V0t )
& ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V2s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V0t ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V1f @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V2s @ V0t ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CASES,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o,V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V3s )
& ( c_2Ereal__topology_2EClosed @ V4t )
& ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V3s )
& ( c_2Ereal__topology_2Econtinuous__on @ V2g @ V4t )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V3s )
& ( (~) @ ( V0P @ V5x ) ) )
| ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V4t )
& ( V0P @ V5x ) ) )
=> ( ( V1f @ V5x )
= ( V2g @ V5x ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( V0P @ V6x ) @ ( V1f @ V6x ) @ ( V2g @ V6x ) )
@ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V3s @ V4t ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__UNION,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V1s )
& ( c_2Ereal__topology_2EClosed @ V2t )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2t ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V1s @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__WITHIN__UNION,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1t: tyop_2Erealax_2Ereal > $o,V2s: tyop_2Erealax_2Ereal > $o,V3l: tyop_2Erealax_2Ereal,V4f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V4f @ V3l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0x ) @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V2s @ V1t ) ) )
<=> ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V4f @ V3l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0x ) @ V2s ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V4f @ V3l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0x ) @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__MAP__CLOSURES,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
<=> ! [V1s: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__COMPONENT__GE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ! [V4y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1s )
=> ( c_2Ereal_2Ereal__lte @ V3b @ ( V0f @ V4y ) ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) )
=> ( c_2Ereal_2Ereal__lte @ V3b @ ( V0f @ V2x ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__COMPONENT__LE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ! [V4y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V4y ) @ V3b ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V2x ) @ V3b ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__ABS__LE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ! [V4y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V4y ) ) @ V3b ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V2x ) ) @ V3b ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__IMAGE__BOUNDED,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ( c_2Ereal__topology_2Ebounded__def @ V1s ) )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__IMAGE__CLOSURE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) )
= ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2EIMAGE__CLOSURE__SUBSET,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ( c_2Ereal__topology_2EClosed @ V2t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) @ V2t ) ) ).
thf(thm_2Ereal__topology_2ELIM__DROP__LBOUND,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net )
& ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ereal_2Ereal__lte @ V3b @ ( V1f @ V4x ) )
@ V0net ) )
=> ( c_2Ereal_2Ereal__lte @ V3b @ V2l ) ) ).
thf(thm_2Ereal__topology_2ELIM__DROP__UBOUND,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net )
& ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ereal_2Ereal__lte @ ( V1f @ V4x ) @ V3b )
@ V0net ) )
=> ( c_2Ereal_2Ereal__lte @ V2l @ V3b ) ) ).
thf(thm_2Ereal__topology_2ELIM__DROP__LE,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal,V4m: tyop_2Erealax_2Ereal] :
( ( ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l @ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ V4m @ V0net )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V5x: A_27a] : ( c_2Ereal_2Ereal__lte @ ( V1f @ V5x ) @ ( V2g @ V5x ) )
@ V0net ) )
=> ( c_2Ereal_2Ereal__lte @ V3l @ V4m ) ) ).
thf(thm_2Ereal__topology_2ELIM__COMPONENT__LE,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal,V4m: tyop_2Erealax_2Ereal] :
( ( ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l @ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ V4m @ V0net )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V5x: A_27a] : ( c_2Ereal_2Ereal__lte @ ( V1f @ V5x ) @ ( V2g @ V5x ) )
@ V0net ) )
=> ( c_2Ereal_2Ereal__lte @ V3l @ V4m ) ) ).
thf(thm_2Ereal__topology_2ELIM__COMPONENT__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2i: A_27b,V3l: tyop_2Erealax_2Ereal,V4b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l @ V0net )
& ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V5x: A_27a] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V1f @ V5x ) @ V4b )
@ V0net ) )
=> ( V3l = V4b ) ) ).
thf(thm_2Ereal__topology_2ELIM__COMPONENT__LBOUND,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ereal_2Ereal__lte @ V3b @ ( V1f @ V4x ) )
@ V0net ) )
=> ( c_2Ereal_2Ereal__lte @ V3b @ V2l ) ) ).
thf(thm_2Ereal__topology_2ELIM__COMPONENT__UBOUND,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal,V4k: A_27b] :
( ( ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V5x: A_27a] : ( c_2Ereal_2Ereal__lte @ ( V1f @ V5x ) @ V3b )
@ V0net ) )
=> ( c_2Ereal_2Ereal__lte @ V2l @ V3b ) ) ).
thf(thm_2Ereal__topology_2ECARD__FRONTIER__INTERVAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eis__interval @ V0s )
=> ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ V0s ) )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EIS__INTERVAL__SCALING__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eis__interval
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V1c @ V2x )
@ V0s ) )
<=> ( ( V1c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( c_2Ereal__topology_2Eis__interval @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EIS__INTERVAL__SCALING,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eis__interval @ V0s )
=> ( c_2Ereal__topology_2Eis__interval
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V1c @ V2x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EIS__INTERVAL__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eis__interval @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EINTERVAL__CONTAINS__COMPACT__NEIGHBOURHOOD,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Eis__interval @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) )
=> ? [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal,V4d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4d )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V4d ) ) @ V0s ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERVAL__SUBSET__IS__INTERVAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eis__interval @ V0s )
=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ V0s )
<=> ( ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2b @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EIS__INTERVAL__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Eis__interval @ V0s )
& ( c_2Ereal__topology_2Eis__interval @ V1t ) )
=> ( c_2Ereal__topology_2Eis__interval @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EIS__INTERVAL__CASES,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eis__interval @ V0s )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( V0s
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
| ? [V1a: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Erealax_2Ereal__lt @ V1a @ V2x ) ) ) )
| ? [V3a: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ereal_2Ereal__lte @ V3a @ V4x ) ) ) )
| ? [V5b: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ereal_2Ereal__lte @ V6x @ V5b ) ) ) )
| ? [V7b: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Erealax_2Ereal__lt @ V8x @ V7b ) ) ) )
| ? [V9a: tyop_2Erealax_2Ereal,V10b: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V11x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V11x @ ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ V9a @ V11x ) @ ( c_2Erealax_2Ereal__lt @ V11x @ V10b ) ) ) ) )
| ? [V12a: tyop_2Erealax_2Ereal,V13b: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V14x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V14x @ ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ V12a @ V14x ) @ ( c_2Ereal_2Ereal__lte @ V14x @ V13b ) ) ) ) )
| ? [V15a: tyop_2Erealax_2Ereal,V16b: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V17x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V17x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V15a @ V17x ) @ ( c_2Erealax_2Ereal__lt @ V17x @ V16b ) ) ) ) )
| ? [V18a: tyop_2Erealax_2Ereal,V19b: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V20x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V20x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V18a @ V20x ) @ ( c_2Ereal_2Ereal__lte @ V20x @ V19b ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EIS__INTERVAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eis__interval @ V0s )
<=> ! [V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal,V3x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2b @ V0s )
& ( c_2Ereal_2Ereal__lte @ V1a @ V3x )
& ( c_2Ereal_2Ereal__lte @ V3x @ V2b ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EIS__INTERVAL__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Eis__interval @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V0s ) )
<=> ? [V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EIS__INTERVAL__POINTWISE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eis__interval @ V0s )
=> ( ? [V2a: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2a @ V0s )
& ( V2a = V1x ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EIS__INTERVAL__UNIV,axiom,
c_2Ereal__topology_2Eis__interval @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2EIS__INTERVAL__EMPTY,axiom,
c_2Ereal__topology_2Eis__interval @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2EIS__INTERVAL__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eis__interval @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
& ( c_2Ereal__topology_2Eis__interval @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEGMENT__SCALAR__MULTIPLE,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2v: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0a @ V2v ) @ ( c_2Erealax_2Ereal__mul @ V1b @ V2v ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__mul @ V3x @ V2v ) @ ( c_2Ebool_2E_5C_2F @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V0a @ V3x ) @ ( c_2Ereal_2Ereal__lte @ V3x @ V1b ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V1b @ V3x ) @ ( c_2Ereal_2Ereal__lte @ V3x @ V0a ) ) ) ) ) )
& ! [V4a: tyop_2Erealax_2Ereal,V5b: tyop_2Erealax_2Ereal,V6v: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V6v
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V4a @ V6v ) @ ( c_2Erealax_2Ereal__mul @ V5b @ V6v ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__mul @ V7x @ V6v ) @ ( c_2Ebool_2E_5C_2F @ ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ V4a @ V7x ) @ ( c_2Erealax_2Ereal__lt @ V7x @ V5b ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ V5b @ V7x ) @ ( c_2Erealax_2Ereal__lt @ V7x @ V4a ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__SEGMENT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EOpen @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ).
thf(thm_2Ereal__topology_2ESEGMENT,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal_2Ereal__lte @ V0a @ V1b ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V0a ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal_2Ereal__lte @ V2a @ V3b ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3b @ V2a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERVAL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal_2Ereal__lte @ V0a @ V1b ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Erealax_2Ereal__lt @ V2a @ V3b ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__DIFF__OPEN__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__INTERVAL__0,axiom,
! [V0e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0e ) )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V0e ) @ V0e ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBALL__INTERVAL__0,axiom,
! [V0e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0e ) )
= ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V0e ) @ V0e ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__INTERVAL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ V0x @ V1e ) @ ( c_2Erealax_2Ereal__add @ V0x @ V1e ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBALL__INTERVAL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) )
= ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ V0x @ V1e ) @ ( c_2Erealax_2Ereal__add @ V0x @ V1e ) ) ) ) ).
thf(thm_2Ereal__topology_2EFINITE__INTERVAL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ereal_2Ereal__lte @ V1b @ V0a ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) )
= ( c_2Ereal_2Ereal__lte @ V3b @ V2a ) ) ) ).
thf(thm_2Ereal__topology_2EFINITE__SPHERE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) ) ).
thf(thm_2Ereal__topology_2ESPHERE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Erealax_2Ereal__lt @ V1r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ V0a @ V1r ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V0a @ V1r ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1r ) )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ V0x @ V1r ) @ ( c_2Erealax_2Ereal__add @ V0x @ V1r ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1r ) )
= ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ V0x @ V1r ) @ ( c_2Erealax_2Ereal__add @ V0x @ V1r ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__OPEN__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0a @ V1b )
=> ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__CLOSED__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERVAL__CASES,axiom,
! [V0b: tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V0b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V0b ) ) )
| ( V2x = V1a )
| ( V2x = V0b ) ) ) ).
thf(thm_2Ereal__topology_2EEQ__BALLS,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1a_27: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal,V3r_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2r ) )
= ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a_27 @ V3r_27 ) ) )
<=> ( ( ( V0a = V1a_27 )
& ( V2r = V3r_27 ) )
| ( ( c_2Ereal_2Ereal__lte @ V2r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( c_2Ereal_2Ereal__lte @ V3r_27 @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) )
& ! [V4a: tyop_2Erealax_2Ereal,V5a_27: tyop_2Erealax_2Ereal,V6r: tyop_2Erealax_2Ereal,V7r_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V6r ) )
= ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5a_27 @ V7r_27 ) ) )
<=> ( ( c_2Ereal_2Ereal__lte @ V6r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( c_2Erealax_2Ereal__lt @ V7r_27 @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
& ! [V8a: tyop_2Erealax_2Ereal,V9a_27: tyop_2Erealax_2Ereal,V10r: tyop_2Erealax_2Ereal,V11r_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V10r ) )
= ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V9a_27 @ V11r_27 ) ) )
<=> ( ( c_2Erealax_2Ereal__lt @ V10r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( c_2Ereal_2Ereal__lte @ V11r_27 @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
& ! [V12a: tyop_2Erealax_2Ereal,V13a_27: tyop_2Erealax_2Ereal,V14r: tyop_2Erealax_2Ereal,V15r_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V12a @ V14r ) )
= ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V13a_27 @ V15r_27 ) ) )
<=> ( ( ( V12a = V13a_27 )
& ( V14r = V15r_27 ) )
| ( ( c_2Erealax_2Ereal__lt @ V14r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( c_2Erealax_2Ereal__lt @ V15r_27 @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__INTERVAL__EQ,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecompact @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecompact @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) )
<=> ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__INTERVAL__EQ,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOpen @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EOpen @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__INTERVAL__EQ,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EClosed @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EClosed @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) )
<=> ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EEQ__INTERVAL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ( ( ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
| ( ( V0a = V2c )
& ( V1b = V3d ) ) ) )
& ! [V4a: tyop_2Erealax_2Ereal,V5b: tyop_2Erealax_2Ereal,V6c: tyop_2Erealax_2Ereal,V7d: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V5b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6c @ V7d ) ) )
<=> ( ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V5b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6c @ V7d ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
& ! [V8a: tyop_2Erealax_2Ereal,V9b: tyop_2Erealax_2Ereal,V10c: tyop_2Erealax_2Ereal,V11d: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V9b ) )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V10c @ V11d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ( ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V9b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V10c @ V11d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
& ! [V12a: tyop_2Erealax_2Ereal,V13b: tyop_2Erealax_2Ereal,V14c: tyop_2Erealax_2Ereal,V15d: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V12a @ V13b ) )
= ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V14c @ V15d ) ) )
<=> ( ( ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V12a @ V13b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V14c @ V15d ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
| ( ( V12a = V14c )
& ( V13b = V15d ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EIMAGE__TWIZZLE__INTERVAL,axiom,
! [A_27a: $tType,V0p: A_27a,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : V3x
@ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__INTERVAL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ediameter @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1b @ V0a ) ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ediameter @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V3b @ V2a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__CONTAINS__OPEN__INTERVAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ? [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__CONTAINS__INTERVAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ? [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__CONTAINS__INTERVAL__OPEN__INTERVAL,axiom,
( ! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ? [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ V0s ) ) ) )
& ! [V4s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V4s )
<=> ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V4s )
=> ? [V6a: tyop_2Erealax_2Ereal,V7b: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6a @ V7b ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6a @ V7b ) ) @ V4s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUMS__INTERVALS,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V4x: tyop_2Erealax_2Ereal,V5y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V4x @ V5y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ) )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V0a @ V2c ) @ ( c_2Erealax_2Ereal__add @ V1b @ V3d ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) )
& ! [V6a: tyop_2Erealax_2Ereal,V7b: tyop_2Erealax_2Ereal,V8c: tyop_2Erealax_2Ereal,V9d: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6a @ V7b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8c @ V9d ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V10x: tyop_2Erealax_2Ereal,V11y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V10x @ V11y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10x @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6a @ V7b ) ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V11y @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8c @ V9d ) ) ) ) ) ) )
= ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V6a @ V8c ) @ ( c_2Erealax_2Ereal__add @ V7b @ V9d ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__INTERVAL__IMAGE__UNIT__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V2x )
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] :
( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V4f: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V4f @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ V1b @ V0a ) @ V3x ) ) )
@ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERVAL__IMAGE__STRETCH__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
? [V3u: tyop_2Erealax_2Ereal,V4v: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] :
( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V6f: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V6f @ ( c_2Erealax_2Ereal__mul @ ( V2m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V5x ) ) )
@ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3u @ V4v ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EIMAGE__STRETCH__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] :
( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V4f: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V4f @ ( c_2Erealax_2Ereal__mul @ ( V2m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V3x ) ) )
@ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal )
@ ( c_2Ereal__topology_2ECLOSED__interval
@ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V5f: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V5f @ ( c_2Ereal_2Emin @ ( c_2Erealax_2Ereal__mul @ ( V2m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a ) @ ( c_2Erealax_2Ereal__mul @ ( V2m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1b ) ) ) )
@ ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V6f: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V6f @ ( c_2Ereal_2Emax @ ( c_2Erealax_2Ereal__mul @ ( V2m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a ) @ ( c_2Erealax_2Ereal__mul @ ( V2m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1b ) ) ) ) )
@ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EUNIT__INTERVAL__NONEMPTY,axiom,
( ( (~)
@ ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EEMPTY__AS__INTERVAL,axiom,
( ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERVAL__TRANSLATION,axiom,
( ! [V0c: tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V0c @ V1a ) @ ( c_2Erealax_2Ereal__add @ V0c @ V2b ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0c @ V3x )
@ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) )
& ! [V4c: tyop_2Erealax_2Ereal,V5a: tyop_2Erealax_2Ereal,V6b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V4c @ V5a ) @ ( c_2Erealax_2Ereal__add @ V4c @ V6b ) ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V4c @ V7x )
@ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5a @ V6b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTER__INTERVAL__MIXED__EQ__EMPTY,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__OPEN__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Efrontier @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__CLOSED__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Efrontier @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SUBSET__CLOSED__INTERVAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ? [V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SUBSET__CLOSED__INTERVAL__SYMMETRIC,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ? [V1a: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V1a ) @ V1a ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SUBSET__OPEN__INTERVAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ? [V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SUBSET__OPEN__INTERVAL__SYMMETRIC,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ? [V1a: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V1a ) @ V1a ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__INTERVAL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__OPEN__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__INV__LE__1,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Einv @ V0x ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LE__INV2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Einv @ V1y ) @ ( c_2Erealax_2Einv @ V0x ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LT__INV2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ V1y ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Einv @ V1y ) @ ( c_2Erealax_2Einv @ V0x ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__POW__LE__1,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1x )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ V1x @ V0n ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__POW__1__LE,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
& ( c_2Ereal_2Ereal__lte @ V1x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Epow @ V1x @ V0n ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBIGUNION__BIGINTER,axiom,
! [A_27a: $tType,V0s: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27a @ V0s )
= ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a )
@ ( c_2Epred__set_2EBIGINTER @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V1t: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V1t ) @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1t @ V0s ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBIGINTER__BIGUNION,axiom,
! [A_27a: $tType,V0s: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EBIGINTER @ A_27a @ V0s )
= ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a )
@ ( c_2Epred__set_2EBIGUNION @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V1t: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V1t ) @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1t @ V0s ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIFF__BIGINTER,axiom,
! [A_27a: $tType,V0u: A_27a > $o,V1s: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EDIFF @ A_27a @ V0u @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1s ) )
= ( c_2Epred__set_2EBIGUNION @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V2t: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ ( c_2Epred__set_2EDIFF @ A_27a @ V0u @ V2t ) @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ V1s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSET__ANTISYM__EQ,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s ) )
<=> ( V0s = V1t ) ) ).
thf(thm_2Ereal__topology_2EFUN__IN__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V2x ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LT__POW2,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0n ) ) ).
thf(thm_2Ereal__topology_2ESUBSET__BIGUNION,axiom,
! [A_27a: $tType,V0f: ( A_27a > $o ) > $o,V1g: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V0f @ V1g )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0f ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1g ) ) ) ).
thf(thm_2Ereal__topology_2ESIMPLE__IMAGE__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1P: A_27a > $o] :
( ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V0f @ V2x ) @ ( V1P @ V2x ) ) )
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( V1P @ V3x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LT__LCANCEL__IMP,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) ) )
=> ( c_2Erealax_2Ereal__lt @ V1y @ V2z ) ) ).
thf(thm_2Ereal__topology_2EPOWERSET__CLAUSES,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( ( c_2Epred__set_2EGSPEC @ ( A_27b > $o ) @ ( A_27b > $o )
@ ^ [V0s: A_27b > $o] : ( c_2Epair_2E_2C @ ( A_27b > $o ) @ $o @ V0s @ ( c_2Epred__set_2ESUBSET @ A_27b @ V0s @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) )
= ( c_2Epred__set_2EINSERT @ ( A_27b > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) @ ( c_2Epred__set_2EEMPTY @ ( A_27b > $o ) ) ) )
& ! [V1a: A_27a,V2t: A_27a > $o] :
( ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V3s: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V3s @ ( c_2Epred__set_2ESUBSET @ A_27a @ V3s @ ( c_2Epred__set_2EINSERT @ A_27a @ V1a @ V2t ) ) ) )
= ( c_2Epred__set_2EUNION @ ( A_27a > $o )
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V4s: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V4s @ ( c_2Epred__set_2ESUBSET @ A_27a @ V4s @ V2t ) ) )
@ ( c_2Epred__set_2EIMAGE @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V5s: A_27a > $o] : ( c_2Epred__set_2EINSERT @ A_27a @ V1a @ V5s )
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V6s: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V6s @ ( c_2Epred__set_2ESUBSET @ A_27a @ V6s @ V2t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__OF__NUM__GE,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__ge @ ( c_2Ereal_2Ereal__of__num @ V0m ) @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
= ( c_2Earithmetic_2E_3E_3D @ V0m @ V1n ) ) ).
thf(thm_2Ereal__topology_2EABS__LE__0,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2ENOT__EQ,axiom,
! [A_27a: $tType,V0a: A_27a,V1b: A_27a] :
( ( (~) @ ( V0a = V1b ) )
<=> ( (~) @ ( V0a = V1b ) ) ) ).
thf(thm_2Ereal__topology_2ELAMBDA__PAIR,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27a > A_27b > A_27c] :
( ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c
@ ^ [V1x: A_27a,V2y: A_27b] : ( V0P @ V1x @ V2y ) )
= ( ^ [V3p: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( V0P @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V3p ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V3p ) ) ) ) ).
thf(thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LE,axiom,
! [V0R: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( ( ! [V1x: tyop_2Enum_2Enum] : ( V0R @ V1x @ V1x )
& ! [V2x: tyop_2Enum_2Enum,V3y: tyop_2Enum_2Enum,V4z: tyop_2Enum_2Enum] :
( ( ( V0R @ V2x @ V3y )
& ( V0R @ V3y @ V4z ) )
=> ( V0R @ V2x @ V4z ) )
& ! [V5n: tyop_2Enum_2Enum] : ( V0R @ V5n @ ( c_2Enum_2ESUC @ V5n ) ) )
=> ! [V6m: tyop_2Enum_2Enum,V7n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V6m @ V7n )
=> ( V0R @ V6m @ V7n ) ) ) ).
thf(thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LE__EQ,axiom,
! [V0R: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( ( ! [V1x: tyop_2Enum_2Enum] : ( V0R @ V1x @ V1x )
& ! [V2x: tyop_2Enum_2Enum,V3y: tyop_2Enum_2Enum,V4z: tyop_2Enum_2Enum] :
( ( ( V0R @ V2x @ V3y )
& ( V0R @ V3y @ V4z ) )
=> ( V0R @ V2x @ V4z ) ) )
=> ( ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V5m @ V6n )
=> ( V0R @ V5m @ V6n ) )
<=> ! [V7n: tyop_2Enum_2Enum] : ( V0R @ V7n @ ( c_2Enum_2ESUC @ V7n ) ) ) ) ).
thf(thm_2Ereal__topology_2EADD__SUBR,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) )
= c_2Enum_2E0 ) ).
thf(thm_2Ereal__topology_2EADD__SUBR2,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ V0m @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) )
= c_2Enum_2E0 ) ).
thf(thm_2Ereal__topology_2EADD__SUB2,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) @ V0m )
= V1n ) ).
thf(thm_2Ereal__topology_2ELE__ADDR,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ V1n @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) ) ).
thf(thm_2Ereal__topology_2ELE__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ V0m @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) ) ).
thf(thm_2Ereal__topology_2EFINITE__POWERSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( c_2Epred__set_2EFINITE @ ( A_27a > $o )
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V1t: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V1t @ ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__INJECTIVE__BOUNDED__BELOW__POS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V1x )
= ( V0f @ V2y ) )
=> ( V1x = V2y ) ) )
=> ? [V3B: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3B )
& ! [V4x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V4x ) @ V3B ) @ ( c_2Ereal_2Eabs @ ( V0f @ V4x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__INVERTIBLE__BOUNDED__BELOW,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Elinear @ V1g )
& ( ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f )
= ( c_2Ecombin_2EI @ tyop_2Erealax_2Ereal ) ) )
=> ? [V2B: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V2B @ ( c_2Ereal_2Eabs @ V3x ) ) @ ( c_2Ereal_2Eabs @ ( V0f @ V3x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__INVERTIBLE__BOUNDED__BELOW__POS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Elinear @ V1g )
& ( ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f )
= ( c_2Ecombin_2EI @ tyop_2Erealax_2Ereal ) ) )
=> ? [V2B: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2B )
& ! [V3x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V2B @ ( c_2Ereal_2Eabs @ V3x ) ) @ ( c_2Ereal_2Eabs @ ( V0f @ V3x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__INJECTIVE__IMP__SURJECTIVE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V1x )
= ( V0f @ V2y ) )
=> ( V1x = V2y ) ) )
=> ! [V3y: tyop_2Erealax_2Ereal] :
? [V4x: tyop_2Erealax_2Ereal] :
( ( V0f @ V4x )
= V3y ) ) ).
thf(thm_2Ereal__topology_2ESPAN__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Espan @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2EINDEPENDENT__INJECTIVE__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Eindependent @ V1s )
& ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V2x )
= ( V0f @ V3y ) )
=> ( V2x = V3y ) ) )
=> ( c_2Ereal__topology_2Eindependent @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EINDEPENDENT__INJECTIVE__IMAGE__GEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Eindependent @ V1s )
& ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Espan @ V1s ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ ( c_2Ereal__topology_2Espan @ V1s ) )
& ( ( V0f @ V2x )
= ( V0f @ V3y ) ) )
=> ( V2x = V3y ) ) )
=> ( c_2Ereal__topology_2Eindependent @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EDEPENDENT__EXPLICIT,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Edependent @ V0p )
<=> ? [V1s: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1s @ V0p )
& ? [V3v: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3v @ V1s )
& ( (~)
@ ( ( V2u @ V3v )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
& ( ( c_2Eiterate_2ESum @ tyop_2Erealax_2Ereal @ V1s
@ ^ [V4v: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ ( V2u @ V4v ) @ V4v ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__EXPLICIT,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Espan @ V0p )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1y: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1y
@ ( c_2Ebool_2E_3F @ ( tyop_2Erealax_2Ereal > $o )
@ ^ [V2s: tyop_2Erealax_2Ereal > $o] :
( c_2Ebool_2E_3F @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal )
@ ^ [V3u: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V2s )
@ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2s @ V0p )
@ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal
@ ( c_2Eiterate_2ESum @ tyop_2Erealax_2Ereal @ V2s
@ ^ [V4v: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ ( V3u @ V4v ) @ V4v ) )
@ V1y ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECARD__GE__DIM__INDEPENDENT,axiom,
! [V0v: tyop_2Erealax_2Ereal > $o,V1b: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1b @ V0v )
& ( c_2Ereal__topology_2Eindependent @ V1b )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Ereal__topology_2Edim @ V0v ) @ ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V1b ) ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0v @ ( c_2Ereal__topology_2Espan @ V1b ) ) ) ).
thf(thm_2Ereal__topology_2EINDEPENDENT__INSERT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eindependent @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ V1s ) )
= ( c_2Ebool_2ECOND @ $o @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ V1s ) @ ( c_2Ereal__topology_2Eindependent @ V1s ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal__topology_2Eindependent @ V1s ) @ ( c_2Ebool_2E_7E @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal__topology_2Espan @ V1s ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EIN__SPAN__INSERT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ V2s ) ) )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal__topology_2Espan @ V2s ) ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ V2s ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__BREAKDOWN,axiom,
! [V0b: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0b @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2a @ ( c_2Ereal__topology_2Espan @ V1s ) ) )
=> ? [V3k: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ V2a @ ( c_2Erealax_2Ereal__mul @ V3k @ V0b ) ) @ ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EDELETE @ tyop_2Erealax_2Ereal @ V1s @ V0b ) ) ) ) ).
thf(thm_2Ereal__topology_2EINDEPENDENT__MONO,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Eindependent @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
=> ( c_2Ereal__topology_2Eindependent @ V0s ) ) ).
thf(thm_2Ereal__topology_2EINDEPENDENT__NONZERO,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eindependent @ V0s )
=> ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EMIDPOINT__COLLINEAR,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V0a = V2c ) )
=> ( ( V1b
= ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2c ) ) )
<=> ( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2c @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V2c ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__MIDPOINT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EMIDPOINT__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V1a ) @ ( V0f @ V2b ) ) )
= ( V0f @ ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBETWEEN__MIDPOINT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebetween @ ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
& ( c_2Ereal__topology_2Ebetween @ ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V0a ) ) ) ).
thf(thm_2Ereal__topology_2EMIDPOINT__EQ__ENDPOINT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( ( ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= V0a )
<=> ( V0a = V1b ) )
& ( ( ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= V1b )
<=> ( V0a = V1b ) )
& ( ( V0a
= ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
<=> ( V0a = V1b ) )
& ( ( V1b
= ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
<=> ( V0a = V1b ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__MIDPOINT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) )
= ( c_2Ereal_2E_2F @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) )
= ( c_2Ereal_2E_2F @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ V0a ) )
= ( c_2Ereal_2E_2F @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ V1b ) )
= ( c_2Ereal_2E_2F @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EMIDPOINT__SYM,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V0a ) ) ) ).
thf(thm_2Ereal__topology_2EMIDPOINT__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V0x ) )
= V0x ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__1,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Ereal__topology_2Ecollinear @ V0s ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__DIST__BETWEEN,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V0a ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V1b ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) )
=> ( c_2Ereal__topology_2Ebetween @ V2x @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__BETWEEN__CASES,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2c @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ( ( c_2Ereal__topology_2Ebetween @ V0a @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V2c ) )
| ( c_2Ereal__topology_2Ebetween @ V1b @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V0a ) )
| ( c_2Ereal__topology_2Ebetween @ V2c @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2EBETWEEN__IMP__COLLINEAR,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebetween @ V2x @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
=> ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBETWEEN__ABS,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebetween @ V2x @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
<=> ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V2x @ V0a ) ) @ ( c_2Ereal_2Ereal__sub @ V1b @ V2x ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1b @ V2x ) ) @ ( c_2Ereal_2Ereal__sub @ V2x @ V0a ) ) ) ) ).
thf(thm_2Ereal__topology_2EBETWEEN__TRANS__2,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebetween @ V0a @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V2c ) )
& ( c_2Ereal__topology_2Ebetween @ V3d @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
=> ( c_2Ereal__topology_2Ebetween @ V0a @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) ) ) ).
thf(thm_2Ereal__topology_2EBETWEEN__TRANS,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebetween @ V0a @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V2c ) )
& ( c_2Ereal__topology_2Ebetween @ V3d @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2c ) ) )
=> ( c_2Ereal__topology_2Ebetween @ V3d @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V2c ) ) ) ).
thf(thm_2Ereal__topology_2EBETWEEN__ANTISYM,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebetween @ V0a @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V2c ) )
& ( c_2Ereal__topology_2Ebetween @ V1b @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2c ) ) )
=> ( V0a = V1b ) ) ).
thf(thm_2Ereal__topology_2EBETWEEN__SYM,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebetween @ V2x @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Ereal__topology_2Ebetween @ V2x @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V0a ) ) ) ).
thf(thm_2Ereal__topology_2EBETWEEN__REFL__EQ,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebetween @ V1x @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V0a ) )
<=> ( V1x = V0a ) ) ).
thf(thm_2Ereal__topology_2EBETWEEN__REFL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebetween @ V0a @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
& ( c_2Ereal__topology_2Ebetween @ V1b @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
& ( c_2Ereal__topology_2Ebetween @ V0a @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V0a ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__3__TRANS,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2c @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2c @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3d @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( (~) @ ( V1b = V2c ) ) )
=> ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3d @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__4__3,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V0a = V1b ) )
=> ( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2c @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3d @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) )
<=> ( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2c @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3d @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__TRIPLES,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V1a = V2b ) )
=> ( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2b @ V0s ) ) )
<=> ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2b @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__3__EXPAND,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2c @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ( ( V0a = V2c )
| ? [V3u: tyop_2Erealax_2Ereal] :
( V1b
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V3u @ V0a ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V3u ) @ V2c ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EMUL__CAUCHY__SCHWARZ__EQUAL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__mul @ V0x @ V0x ) @ ( c_2Erealax_2Ereal__mul @ V1y @ V1y ) ) )
<=> ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__EQUAL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) )
<=> ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__LEMMA__ALT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ( ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ? [V2c: tyop_2Erealax_2Ereal] :
( V1y
= ( c_2Erealax_2Ereal__mul @ V2c @ V0x ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__LEMMA,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ( ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( V1y
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ? [V2c: tyop_2Erealax_2Ereal] :
( V1y
= ( c_2Erealax_2Ereal__mul @ V2c @ V0x ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__3,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2z @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ V2z @ V1y ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__SMALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V0s ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( c_2Ereal__topology_2Ecollinear @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__SING,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__EMPTY,axiom,
c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecollinear @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
=> ( c_2Ereal__topology_2Ecollinear @ V0s ) ) ).
thf(thm_2Ereal__topology_2EDIST__TRIANGLE__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2z ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1y @ V2z ) ) ) )
<=> ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) @ ( c_2Ereal_2Ereal__sub @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1y @ V2z ) ) @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) ) ) ).
thf(thm_2Ereal__topology_2EABS__TRIANGLE__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) )
<=> ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V0x ) @ V1y )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V1y ) @ V0x ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__EQ__RINV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( V0x
= ( c_2Erealax_2Ereal__neg @ V0x ) )
<=> ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__EQ__LINV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__neg @ V0x )
= V0x )
<=> ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__ABS__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) )
<=> ( ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V0x ) @ V1y )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V1y ) @ V0x ) )
| ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V0x ) @ V1y )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Eabs @ V1y ) ) @ V0x ) ) ) ) ).
thf(thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) )
<=> ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V0x ) @ V1y )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V1y ) @ V0x ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Espan @ V0s ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ ( c_2Ereal__topology_2Espan @ V1t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__SUMS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Esubspace @ V0s )
& ( c_2Ereal__topology_2Esubspace @ V1t ) )
=> ( c_2Ereal__topology_2Esubspace
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__TRANSLATION__SELF__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esubspace @ V0s )
=> ( ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V1a @ V2x )
@ V0s )
= V0s )
<=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__TRANSLATION__SELF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Esubspace @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V0s ) )
=> ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V1a @ V2x )
@ V0s )
= V0s ) ) ).
thf(thm_2Ereal__topology_2ESURJECTIVE__IMAGE__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ! [V3y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3y @ V2t )
=> ? [V4x: A_27a] :
( ( V0f @ V4x )
= V3y ) )
& ! [V5x: A_27a] :
( ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V5x ) @ V2t )
= ( c_2Ebool_2EIN @ A_27a @ V5x @ V1s ) ) )
=> ( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s )
= V2t ) ) ).
thf(thm_2Ereal__topology_2ESPAN__SUBSET__SUBSPACE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ( c_2Ereal__topology_2Esubspace @ V1t ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Espan @ V0s ) @ V1t ) ) ).
thf(thm_2Ereal__topology_2ESPAN__EQ__SELF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Espan @ V0s )
= V0s )
<=> ( c_2Ereal__topology_2Esubspace @ V0s ) ) ).
thf(thm_2Ereal__topology_2ESPAN__ADD__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2Espan @ V0s ) )
=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V1x @ V2y ) @ ( c_2Ereal__topology_2Espan @ V0s ) )
= ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ ( c_2Ereal__topology_2Espan @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__SUM,axiom,
! [A_27a: $tType,V0s: tyop_2Erealax_2Ereal > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2t )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2t )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1f @ V3x ) @ ( c_2Ereal__topology_2Espan @ V0s ) ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Eiterate_2ESum @ A_27a @ V2t @ V1f ) @ ( c_2Ereal__topology_2Espan @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Espan @ V2s ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal__topology_2Espan @ V2s ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) @ ( c_2Ereal__topology_2Espan @ V2s ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__NEG__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V0x ) @ ( c_2Ereal__topology_2Espan @ V1s ) )
= ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Espan @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Espan @ V1s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V0x ) @ ( c_2Ereal__topology_2Espan @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__MUL__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( (~)
@ ( V1c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V1c @ V0x ) @ ( c_2Ereal__topology_2Espan @ V2s ) )
= ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Espan @ V2s ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Espan @ V2s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V1c @ V0x ) @ ( c_2Ereal__topology_2Espan @ V2s ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Espan @ V2s ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal__topology_2Espan @ V2s ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ ( c_2Ereal__topology_2Espan @ V2s ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__0,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2Espan @ V0s ) ) ).
thf(thm_2Ereal__topology_2ESPAN__UNIV,axiom,
( ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ESPAN__UNION__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Espan @ V0s ) @ ( c_2Ereal__topology_2Espan @ V1t ) ) @ ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__INC,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Espan @ V0s ) ) ).
thf(thm_2Ereal__topology_2ESPAN__SUPERSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2Espan @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__INDUCT__ALT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1h: tyop_2Erealax_2Ereal > $o] :
( ( ( V1h @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ! [V2c: tyop_2Erealax_2Ereal,V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
& ( V1h @ V4y ) )
=> ( V1h @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V2c @ V3x ) @ V4y ) ) ) )
=> ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ ( c_2Ereal__topology_2Espan @ V0s ) )
=> ( V1h @ V5x ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__SUBSPACE,axiom,
! [V0b: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0b @ V1s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1s @ ( c_2Ereal__topology_2Espan @ V0b ) )
& ( c_2Ereal__topology_2Esubspace @ V1s ) )
=> ( ( c_2Ereal__topology_2Espan @ V0b )
= V1s ) ) ).
thf(thm_2Ereal__topology_2EDEPENDENT__MONO,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Edependent @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
=> ( c_2Ereal__topology_2Edependent @ V1t ) ) ).
thf(thm_2Ereal__topology_2EREAL__LE__SQUARE__ABS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ V1y @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__EQ__SQUARE__ABS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Eabs @ V0x )
= ( c_2Ereal_2Eabs @ V1y ) )
<=> ( ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Ereal_2Epow @ V1y @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__HALF,axiom,
( ! [V0e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ V0e @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0e ) )
& ! [V1e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2E_2F @ V1e @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2E_2F @ V1e @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
= V1e )
& ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2E_2F @ V2e @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
= V2e ) ) ).
thf(thm_2Ereal__topology_2EFINITE__SUBSET__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V2t )
& ( c_2Epred__set_2ESUBSET @ A_27b @ V2t @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) ) )
<=> ? [V3s_27: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V3s_27 )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3s_27 @ V1s )
& ( V2t
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V3s_27 ) ) ) ) ).
thf(thm_2Ereal__topology_2EEXISTS__FINITE__SUBSET__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( A_27a > $o ) > $o,V1f: A_27b > A_27a,V2s: A_27b > $o] :
( ? [V3t: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V3t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3t @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ V1f @ V2s ) )
& ( V0P @ V3t ) )
<=> ? [V4t: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ V4t )
& ( c_2Epred__set_2ESUBSET @ A_27b @ V4t @ V2s )
& ( V0P @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ V1f @ V4t ) ) ) ) ).
thf(thm_2Ereal__topology_2EFORALL__FINITE__SUBSET__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( A_27a > $o ) > $o,V1f: A_27b > A_27a,V2s: A_27b > $o] :
( ! [V3t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V3t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3t @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ V1f @ V2s ) ) )
=> ( V0P @ V3t ) )
<=> ! [V4t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V4t )
& ( c_2Epred__set_2ESUBSET @ A_27b @ V4t @ V2s ) )
=> ( V0P @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ V1f @ V4t ) ) ) ) ).
thf(thm_2Ereal__topology_2EFORALL__IN__GSPEC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,V0Q: A_27b > $o] :
( ! [V1P: A_27a > $o,V2f: A_27a > A_27b] :
( ! [V3z: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3z
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V2f @ V4x ) @ ( V1P @ V4x ) ) ) )
=> ( V0Q @ V3z ) )
<=> ! [V5x: A_27a] :
( ( V1P @ V5x )
=> ( V0Q @ ( V2f @ V5x ) ) ) )
& ! [V6P: A_27c > A_27d > $o,V7f: A_27c > A_27d > A_27b] :
( ! [V8z: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V8z
@ ( c_2Epred__set_2EGSPEC @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d )
@ ( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ ( tyop_2Epair_2Eprod @ A_27b @ $o )
@ ^ [V9x: A_27c,V10y: A_27d] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V7f @ V9x @ V10y ) @ ( V6P @ V9x @ V10y ) ) ) ) )
=> ( V0Q @ V8z ) )
<=> ! [V11x: A_27c,V12y: A_27d] :
( ( V6P @ V11x @ V12y )
=> ( V0Q @ ( V7f @ V11x @ V12y ) ) ) )
& ! [V13P: A_27e > A_27f > A_27g > $o,V14f: A_27e > A_27f > A_27g > A_27b] :
( ! [V15z: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V15z
@ ( c_2Epred__set_2EGSPEC @ A_27b @ ( tyop_2Epair_2Eprod @ A_27e @ ( tyop_2Epair_2Eprod @ A_27f @ A_27g ) )
@ ( c_2Epair_2EUNCURRY @ A_27e @ ( tyop_2Epair_2Eprod @ A_27f @ A_27g ) @ ( tyop_2Epair_2Eprod @ A_27b @ $o )
@ ^ [V16w: A_27e] :
( c_2Epair_2EUNCURRY @ A_27f @ A_27g @ ( tyop_2Epair_2Eprod @ A_27b @ $o )
@ ^ [V17x: A_27f,V18y: A_27g] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V14f @ V16w @ V17x @ V18y ) @ ( V13P @ V16w @ V17x @ V18y ) ) ) ) ) )
=> ( V0Q @ V15z ) )
<=> ! [V19w: A_27e,V20x: A_27f,V21y: A_27g] :
( ( V13P @ V19w @ V20x @ V21y )
=> ( V0Q @ ( V14f @ V19w @ V20x @ V21y ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EEXISTS__IN__GSPEC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,V0Q: A_27b > $o] :
( ! [V1P: A_27a > $o,V2f: A_27a > A_27b] :
( ? [V3z: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3z
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V2f @ V4x ) @ ( V1P @ V4x ) ) ) )
& ( V0Q @ V3z ) )
<=> ? [V5x: A_27a] :
( ( V1P @ V5x )
& ( V0Q @ ( V2f @ V5x ) ) ) )
& ! [V6P: A_27c > A_27d > $o,V7f: A_27c > A_27d > A_27b] :
( ? [V8z: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V8z
@ ( c_2Epred__set_2EGSPEC @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d )
@ ( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ ( tyop_2Epair_2Eprod @ A_27b @ $o )
@ ^ [V9x: A_27c,V10y: A_27d] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V7f @ V9x @ V10y ) @ ( V6P @ V9x @ V10y ) ) ) ) )
& ( V0Q @ V8z ) )
<=> ? [V11x: A_27c,V12y: A_27d] :
( ( V6P @ V11x @ V12y )
& ( V0Q @ ( V7f @ V11x @ V12y ) ) ) )
& ! [V13P: A_27e > A_27f > A_27g > $o,V14f: A_27e > A_27f > A_27g > A_27b] :
( ? [V15z: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V15z
@ ( c_2Epred__set_2EGSPEC @ A_27b @ ( tyop_2Epair_2Eprod @ A_27e @ ( tyop_2Epair_2Eprod @ A_27f @ A_27g ) )
@ ( c_2Epair_2EUNCURRY @ A_27e @ ( tyop_2Epair_2Eprod @ A_27f @ A_27g ) @ ( tyop_2Epair_2Eprod @ A_27b @ $o )
@ ^ [V16w: A_27e] :
( c_2Epair_2EUNCURRY @ A_27f @ A_27g @ ( tyop_2Epair_2Eprod @ A_27b @ $o )
@ ^ [V17x: A_27f,V18y: A_27g] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V14f @ V16w @ V17x @ V18y ) @ ( V13P @ V16w @ V17x @ V18y ) ) ) ) ) )
& ( V0Q @ V15z ) )
<=> ? [V19w: A_27e,V20x: A_27f,V21y: A_27g] :
( ( V13P @ V19w @ V20x @ V21y )
& ( V0Q @ ( V14f @ V19w @ V20x @ V21y ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EEMPTY__BIGUNION,axiom,
! [A_27a: $tType,V0s: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2EBIGUNION @ A_27a @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ! [V1t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1t @ V0s )
=> ( V1t
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LT__RNEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Erealax_2Ereal__neg @ V1y ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2EUPPER__BOUND__FINITE__SET,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ? [V2a: tyop_2Enum_2Enum] :
! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V0f @ V3x ) @ V2a ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__BOUNDS__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1k: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ V1k ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ V1k ) )
<=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V0x ) @ V1k ) ) ).
thf(thm_2Ereal__topology_2EBIGUNION__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( A_27b > $o ) @ V0f @ V1s ) )
= ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V2y: A_27b] :
( c_2Epair_2E_2C @ A_27b @ $o @ V2y
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ A_27b @ V2y @ ( V0f @ V3x ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBIGINTER__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2EBIGINTER @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( A_27b > $o ) @ V0f @ V1s ) )
= ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V2y: A_27b] :
( c_2Epair_2E_2C @ A_27b @ $o @ V2y
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V3x: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ A_27b @ V2y @ ( V0f @ V3x ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LE__LMUL1,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V1y @ V2z ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) ) ) ).
thf(thm_2Ereal__topology_2ELE__EXISTS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
<=> ? [V2d: tyop_2Enum_2Enum] :
( V1n
= ( c_2Earithmetic_2E_2B @ V0m @ V2d ) ) ) ).
thf(thm_2Ereal__topology_2ELT__EXISTS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
<=> ? [V2d: tyop_2Enum_2Enum] :
( V1n
= ( c_2Earithmetic_2E_2B @ V0m @ ( c_2Enum_2ESUC @ V2d ) ) ) ) ).
thf(thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LT__EQ,axiom,
! [V0R: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( ! [V1x: tyop_2Enum_2Enum,V2y: tyop_2Enum_2Enum,V3z: tyop_2Enum_2Enum] :
( ( ( V0R @ V1x @ V2y )
& ( V0R @ V2y @ V3z ) )
=> ( V0R @ V1x @ V3z ) )
=> ( ! [V4m: tyop_2Enum_2Enum,V5n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V4m @ V5n )
=> ( V0R @ V4m @ V5n ) )
<=> ! [V6n: tyop_2Enum_2Enum] : ( V0R @ V6n @ ( c_2Enum_2ESUC @ V6n ) ) ) ) ).
thf(thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LT,axiom,
! [V0R: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( ( ! [V1x: tyop_2Enum_2Enum,V2y: tyop_2Enum_2Enum,V3z: tyop_2Enum_2Enum] :
( ( ( V0R @ V1x @ V2y )
& ( V0R @ V2y @ V3z ) )
=> ( V0R @ V1x @ V3z ) )
& ! [V4n: tyop_2Enum_2Enum] : ( V0R @ V4n @ ( c_2Enum_2ESUC @ V4n ) ) )
=> ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V5m @ V6n )
=> ( V0R @ V5m @ V6n ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__SEQUENTIAL__INJ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elimit__point__of @ V0x @ V1s )
<=> ? [V2f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ! [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V2f @ V3n ) @ ( c_2Epred__set_2EDELETE @ tyop_2Erealax_2Ereal @ V1s @ V0x ) )
& ! [V4m: tyop_2Enum_2Enum,V5n: tyop_2Enum_2Enum] :
( ( ( V2f @ V4m )
= ( V2f @ V5n ) )
<=> ( V4m = V5n ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V2f @ V0x @ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__SEQUENTIAL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elimit__point__of @ V0x @ V1s )
<=> ? [V2f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ! [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V2f @ V3n ) @ ( c_2Epred__set_2EDELETE @ tyop_2Erealax_2Ereal @ V1s @ V0x ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V2f @ V0x @ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2EINFINITE__SUPERSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t ) )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__INFINITE__OPEN__BALL__CBALL,axiom,
( ! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s )
<=> ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V2t )
& ( c_2Ereal__topology_2EOpen @ V2t ) )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V2t ) ) ) ) )
& ! [V3s: tyop_2Erealax_2Ereal > $o,V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elimit__point__of @ V4x @ V3s )
<=> ! [V5e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5e )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V3s @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4x @ V5e ) ) ) ) ) ) )
& ! [V6s: tyop_2Erealax_2Ereal > $o,V7x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elimit__point__of @ V7x @ V6s )
<=> ! [V8e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V8e )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V6s @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V7x @ V8e ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__INFINITE__OPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s )
<=> ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V2t )
& ( c_2Ereal__topology_2EOpen @ V2t ) )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V2t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__INFINITE__BALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2e ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__INFINITE__CBALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2e ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINFINITE__OPEN__IN,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ V1s )
& ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
& ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V0u ) ) )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECONDENSATION__POINT__OF__SUBSET,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econdensation__point__of @ V0x @ V1s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1s @ V2t ) )
=> ( c_2Ereal__topology_2Econdensation__point__of @ V0x @ V2t ) ) ).
thf(thm_2Ereal__topology_2ECONDENSATION__POINT__IMP__LIMPT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econdensation__point__of @ V0x @ V1s )
=> ( c_2Ereal__topology_2Elimit__point__of @ V0x @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__BALL__CBALL,axiom,
( ! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econdensation__point__of @ V1x @ V0s )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ( (~) @ ( c_2Epred__set_2Ecountable @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2e ) ) ) ) ) ) )
& ! [V3s: tyop_2Erealax_2Ereal > $o,V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econdensation__point__of @ V4x @ V3s )
<=> ! [V5e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5e )
=> ( (~) @ ( c_2Epred__set_2Ecountable @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V3s @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4x @ V5e ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__BALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econdensation__point__of @ V1x @ V0s )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ( (~) @ ( c_2Epred__set_2Ecountable @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2e ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__CBALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econdensation__point__of @ V1x @ V0s )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ( (~) @ ( c_2Epred__set_2Ecountable @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2e ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__LINEAR,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2f: A_27a > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2f @ V3l @ V0net )
& ( c_2Ereal__topology_2Elinear @ V1h ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V4x: A_27a] : ( V1h @ ( V2f @ V4x ) )
@ ( V1h @ V3l )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__CONST,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1a: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V2x: A_27a] : V1a
@ V1a
@ V0net ) ).
thf(thm_2Ereal__topology_2ELIM__CMUL,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V4x: A_27a] : ( c_2Erealax_2Ereal__mul @ V3c @ ( V1f @ V4x ) )
@ ( c_2Erealax_2Ereal__mul @ V3c @ V2l )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__CMUL__EQ,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3c: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V3c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V4x: A_27a] : ( c_2Erealax_2Ereal__mul @ V3c @ ( V1f @ V4x ) )
@ ( c_2Erealax_2Ereal__mul @ V3c @ V2l )
@ V0net )
= ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__NEG,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__neg @ ( V1f @ V3x ) )
@ ( c_2Erealax_2Ereal__neg @ V2l )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__NEG__EQ,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__neg @ ( V1f @ V3x ) )
@ ( c_2Erealax_2Ereal__neg @ V2l )
@ V0net )
= ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__ADD,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal,V4m: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l @ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ V4m @ V0net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V5x: A_27a] : ( c_2Erealax_2Ereal__add @ ( V1f @ V5x ) @ ( V2g @ V5x ) )
@ ( c_2Erealax_2Ereal__add @ V3l @ V4m )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__ABS,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ereal_2Eabs @ ( V1f @ V3x ) )
@ ( c_2Ereal_2Eabs @ V2l )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__SUB,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal,V4m: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l @ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ V4m @ V0net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V5x: A_27a] : ( c_2Ereal_2Ereal__sub @ ( V1f @ V5x ) @ ( V2g @ V5x ) )
@ ( c_2Ereal_2Ereal__sub @ V3l @ V4m )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__MAX,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal,V4m: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l @ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ V4m @ V0net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V5x: A_27a] : ( c_2Ereal_2Emax @ ( V1f @ V5x ) @ ( V2g @ V5x ) )
@ ( c_2Ereal_2Emax @ V3l @ V4m )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__MIN,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal,V4m: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l @ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ V4m @ V0net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V5x: A_27a] : ( c_2Ereal_2Emin @ ( V1f @ V5x ) @ ( V2g @ V5x ) )
@ ( c_2Ereal_2Emin @ V3l @ V4m )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__NULL,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net )
= ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ereal_2Ereal__sub @ ( V1f @ V3x ) @ V2l )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__NULL__ABS,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0net )
= ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V2x: A_27a] : ( c_2Ereal_2Eabs @ ( V1f @ V2x ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__NULL__CMUL__EQ,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V2c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__mul @ V2c @ ( V1f @ V3x ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0net )
= ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0net ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__NULL__CMUL,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0net )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__mul @ V2c @ ( V1f @ V3x ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__NULL__ADD,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__add @ ( V1f @ V3x ) @ ( V2g @ V3x ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__NULL__SUB,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ereal_2Ereal__sub @ ( V1f @ V3x ) @ ( V2g @ V3x ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__NULL__COMPARISON,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V1f @ V3x ) ) @ ( V2g @ V3x ) )
@ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V4x: A_27a] : ( V2g @ V4x )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__COMPONENT,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2i: A_27b,V3l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l @ V0net )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V4a: A_27a] : ( V1f @ V4a )
@ V3l
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__TRANSFORM__BOUND,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V3n: A_27a] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V1f @ V3n ) ) @ ( c_2Ereal_2Eabs @ ( V2g @ V3n ) ) )
@ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__NULL__CMUL__BOUNDED,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal,V3B: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V4a: A_27a] : ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V2g @ V4a ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V1f @ V4a ) ) @ V3B ) )
@ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V5n: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V1f @ V5n ) @ ( V2g @ V5n ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__SUM,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27b,V1f: A_27a > A_27b > tyop_2Erealax_2Ereal,V2l: A_27a > tyop_2Erealax_2Ereal,V3s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V3s )
& ! [V4i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4i @ V3s )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27b @ ( V1f @ V4i ) @ ( V2l @ V4i ) @ V0net ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27b
@ ^ [V5x: A_27b] :
( c_2Eiterate_2ESum @ A_27a @ V3s
@ ^ [V6i: A_27a] : ( V1f @ V6i @ V5x ) )
@ ( c_2Eiterate_2ESum @ A_27a @ V3s @ V2l )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__NULL__SUM,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > A_27b > tyop_2Erealax_2Ereal,V2s: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V2s )
& ! [V3a: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3a @ V2s )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V4x: A_27a] : ( V1f @ V4x @ V3a )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0net ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V5x: A_27a] : ( c_2Eiterate_2ESum @ A_27b @ V2s @ ( V1f @ V5x ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__IN__CLOSED__SET,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3l: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V2s )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1f @ V4x ) @ V2s )
@ V0net )
& ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l @ V0net ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3l @ V2s ) ) ).
thf(thm_2Ereal__topology_2ELIM__ABS__UBOUND,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V1f @ V4x ) ) @ V3b )
@ V0net ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V2l ) @ V3b ) ) ).
thf(thm_2Ereal__topology_2ELIM__ABS__LBOUND,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ereal_2Ereal__lte @ V3b @ ( c_2Ereal_2Eabs @ ( V1f @ V4x ) ) )
@ V0net ) )
=> ( c_2Ereal_2Ereal__lte @ V3b @ ( c_2Ereal_2Eabs @ V2l ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__UNIQUE,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3l_27: tyop_2Erealax_2Ereal] :
( ( ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l_27 @ V0net ) )
=> ( V2l = V3l_27 ) ) ).
thf(thm_2Ereal__topology_2ETENDSTO__LIM,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal] :
( ( ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net ) )
=> ( ( c_2Ereal__topology_2Elim__def @ A_27a @ V0net @ V1f )
= V2l ) ) ).
thf(thm_2Ereal__topology_2ELIM__CONST__EQ,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1c: tyop_2Erealax_2Ereal,V2d: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V3x: A_27a] : V1c
@ V2d
@ V0net )
<=> ( ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net )
| ( V1c = V2d ) ) ) ).
thf(thm_2Ereal__topology_2EUNIFORM__LIM__ADD,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1P: A_27b > $o,V2f: A_27b > A_27a > tyop_2Erealax_2Ereal,V3g: A_27b > A_27a > tyop_2Erealax_2Ereal,V4l: A_27b > tyop_2Erealax_2Ereal,V5m: A_27b > tyop_2Erealax_2Ereal] :
( ( ! [V6e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V6e )
=> ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V7x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V8n: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( V1P @ V8n ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V2f @ V8n @ V7x ) @ ( V4l @ V8n ) ) ) @ V6e ) ) )
@ V0net ) )
& ! [V9e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V9e )
=> ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V10x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V11n: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( V1P @ V11n ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V3g @ V11n @ V10x ) @ ( V5m @ V11n ) ) ) @ V9e ) ) )
@ V0net ) ) )
=> ! [V12e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V12e )
=> ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V13x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V14n: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( V1P @ V14n ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__add @ ( V2f @ V14n @ V13x ) @ ( V3g @ V14n @ V13x ) ) @ ( c_2Erealax_2Ereal__add @ ( V4l @ V14n ) @ ( V5m @ V14n ) ) ) ) @ V12e ) ) )
@ V0net ) ) ) ).
thf(thm_2Ereal__topology_2EUNIFORM__LIM__SUB,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1P: A_27b > $o,V2f: A_27b > A_27a > tyop_2Erealax_2Ereal,V3g: A_27b > A_27a > tyop_2Erealax_2Ereal,V4l: A_27b > tyop_2Erealax_2Ereal,V5m: A_27b > tyop_2Erealax_2Ereal] :
( ( ! [V6e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V6e )
=> ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V7x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V8n: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( V1P @ V8n ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V2f @ V8n @ V7x ) @ ( V4l @ V8n ) ) ) @ V6e ) ) )
@ V0net ) )
& ! [V9e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V9e )
=> ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V10x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V11n: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( V1P @ V11n ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V3g @ V11n @ V10x ) @ ( V5m @ V11n ) ) ) @ V9e ) ) )
@ V0net ) ) )
=> ! [V12e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V12e )
=> ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V13x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V14n: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( V1P @ V14n ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__sub @ ( V2f @ V14n @ V13x ) @ ( V3g @ V14n @ V13x ) ) @ ( c_2Ereal_2Ereal__sub @ ( V4l @ V14n ) @ ( V5m @ V14n ) ) ) ) @ V12e ) ) )
@ V0net ) ) ) ).
thf(thm_2Ereal__topology_2EUNIFORM__LIM__BILINEAR,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1P: A_27b > $o,V2h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3f: A_27b > A_27a > tyop_2Erealax_2Ereal,V4g: A_27b > A_27a > tyop_2Erealax_2Ereal,V5l: A_27b > tyop_2Erealax_2Ereal,V6m: A_27b > tyop_2Erealax_2Ereal,V7b1: tyop_2Erealax_2Ereal,V8b2: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebilinear @ V2h )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V9x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V10n: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( V1P @ V10n ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V5l @ V10n ) ) @ V7b1 ) ) )
@ V0net )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V11x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V12n: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( V1P @ V12n ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V6m @ V12n ) ) @ V8b2 ) ) )
@ V0net )
& ! [V13e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V13e )
=> ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V14x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V15n: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( V1P @ V15n ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V3f @ V15n @ V14x ) @ ( V5l @ V15n ) ) ) @ V13e ) ) )
@ V0net ) )
& ! [V16e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V16e )
=> ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V17x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V18n: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( V1P @ V18n ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V4g @ V18n @ V17x ) @ ( V6m @ V18n ) ) ) @ V16e ) ) )
@ V0net ) ) )
=> ! [V19e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V19e )
=> ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V20x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V21n: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( V1P @ V21n ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V2h @ ( V3f @ V21n @ V20x ) @ ( V4g @ V21n @ V20x ) ) @ ( V2h @ ( V5l @ V21n ) @ ( V6m @ V21n ) ) ) ) @ V19e ) ) )
@ V0net ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__BILINEAR,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2f: A_27a > tyop_2Erealax_2Ereal,V3g: A_27a > tyop_2Erealax_2Ereal,V4l: tyop_2Erealax_2Ereal,V5m: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2f @ V4l @ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V3g @ V5m @ V0net )
& ( c_2Ereal__topology_2Ebilinear @ V1h ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V6x: A_27a] : ( V1h @ ( V2f @ V6x ) @ ( V3g @ V6x ) )
@ ( V1h @ V4l @ V5m )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__WITHIN__ID,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : V2x
@ V0a
@ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0a ) @ V1s ) ) ).
thf(thm_2Ereal__topology_2ELIM__AT__ID,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : V1x
@ V0a
@ ( c_2Ereal__topology_2Eat @ V0a ) ) ).
thf(thm_2Ereal__topology_2ELIM__AT__ZERO,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Eat @ V2a ) )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( V0f @ ( c_2Erealax_2Ereal__add @ V2a @ V3x ) )
@ V1l
@ ( c_2Ereal__topology_2Eat @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal__topology_2ENETLIMIT__WITHIN,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0a ) @ V1s ) ) )
=> ( ( c_2Ereal__topology_2Enetlimit @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0a ) @ V1s ) )
= V0a ) ) ).
thf(thm_2Ereal__topology_2ENETLIMIT__AT,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Enetlimit @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0a ) )
= V0a ) ).
thf(thm_2Ereal__topology_2ELIM__TRANSFORM,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ereal_2Ereal__sub @ ( V1f @ V4x ) @ ( V2g @ V4x ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l @ V0net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ V3l @ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__TRANSFORM__EVENTUALLY,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V4x: A_27a] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V1f @ V4x ) @ ( V2g @ V4x ) )
@ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l @ V0net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ V3l @ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN,axiom,
! [V0l: tyop_2Erealax_2Ereal,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3x: tyop_2Erealax_2Ereal,V4s: tyop_2Erealax_2Ereal > $o,V5d: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5d )
& ! [V6x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x_27 @ V4s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6x_27 @ V3x ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6x_27 @ V3x ) ) @ V5d ) )
=> ( ( V1f @ V6x_27 )
= ( V2g @ V6x_27 ) ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V1f @ V0l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3x ) @ V4s ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V2g @ V0l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3x ) @ V4s ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__TRANSFORM__AT,axiom,
! [V0l: tyop_2Erealax_2Ereal,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3x: tyop_2Erealax_2Ereal,V4d: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4d )
& ! [V5x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x_27 @ V3x ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x_27 @ V3x ) ) @ V4d ) )
=> ( ( V1f @ V5x_27 )
= ( V2g @ V5x_27 ) ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V1f @ V0l @ ( c_2Ereal__topology_2Eat @ V3x ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V2g @ V0l @ ( c_2Ereal__topology_2Eat @ V3x ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__TRANSFORM__EQ,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ereal_2Ereal__sub @ ( V1f @ V4x ) @ ( V2g @ V4x ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0net )
=> ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3l @ V0net )
= ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ V3l @ V0net ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__SET,axiom,
! [V0l: tyop_2Erealax_2Ereal,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o,V4t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eeventually @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D @ $o @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V3s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V4t ) )
@ ( c_2Ereal__topology_2Eat @ V2a ) )
=> ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V1f @ V0l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2a ) @ V3s ) )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V1f @ V0l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2a ) @ V4t ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__SET__IMP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o,V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Eeventually @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V4t ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V3s ) )
@ ( c_2Ereal__topology_2Eat @ V2a ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2a ) @ V3s ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2a ) @ V4t ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__TRANSFORM__AWAY__WITHIN,axiom,
! [V0l: tyop_2Erealax_2Ereal,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3a: tyop_2Erealax_2Ereal,V4b: tyop_2Erealax_2Ereal,V5s: tyop_2Erealax_2Ereal > $o] :
( ( ( (~) @ ( V3a = V4b ) )
& ! [V6x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V5s )
& ( (~) @ ( V6x = V3a ) )
& ( (~) @ ( V6x = V4b ) ) )
=> ( ( V1f @ V6x )
= ( V2g @ V6x ) ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V1f @ V0l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3a ) @ V5s ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V2g @ V0l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3a ) @ V5s ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__TRANSFORM__AWAY__AT,axiom,
! [V0l: tyop_2Erealax_2Ereal,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3a: tyop_2Erealax_2Ereal,V4b: tyop_2Erealax_2Ereal] :
( ( ( (~) @ ( V3a = V4b ) )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( ( (~) @ ( V5x = V3a ) )
& ( (~) @ ( V5x = V4b ) ) )
=> ( ( V1f @ V5x )
= ( V2g @ V5x ) ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V1f @ V0l @ ( c_2Ereal__topology_2Eat @ V3a ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V2g @ V0l @ ( c_2Ereal__topology_2Eat @ V3a ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__OPEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3a: tyop_2Erealax_2Ereal,V4l: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EOpen @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3a @ V2s )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V2s )
& ( (~) @ ( V5x = V3a ) ) )
=> ( ( V0f @ V5x )
= ( V1g @ V5x ) ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V4l @ ( c_2Ereal__topology_2Eat @ V3a ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V1g @ V4l @ ( c_2Ereal__topology_2Eat @ V3a ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__OPEN__IN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4a: tyop_2Erealax_2Ereal,V5l: tyop_2Erealax_2Ereal] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4a @ V2s )
& ! [V6x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V2s )
& ( (~) @ ( V6x = V4a ) ) )
=> ( ( V0f @ V6x )
= ( V1g @ V6x ) ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V5l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V4a ) @ V3t ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V1g @ V5l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V4a ) @ V3t ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__CASES__FINITE__SEQUENTIALLY,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum
@ ( c_2Epred__set_2EGSPEC @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ $o @ V4n @ ( V0P @ V4n ) ) ) )
=> ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( V0P @ V5n ) @ ( V1f @ V5n ) @ ( V2g @ V5n ) )
@ V3l
@ c_2Ereal__topology_2Esequentially )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V2g @ V3l @ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__CASES__COFINITE__SEQUENTIALLY,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum
@ ( c_2Epred__set_2EGSPEC @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ $o @ V4n @ ( c_2Ebool_2E_7E @ ( V0P @ V4n ) ) ) ) )
=> ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( V0P @ V5n ) @ ( V1f @ V5n ) @ ( V2g @ V5n ) )
@ V3l
@ c_2Ereal__topology_2Esequentially )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V1f @ V3l @ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__CASES__SEQUENTIALLY,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3m: tyop_2Enum_2Enum] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2E_3C_3D @ V3m @ V4n ) @ ( V0f @ V4n ) @ ( V1g @ V4n ) )
@ V2l
@ c_2Ereal__topology_2Esequentially )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0f @ V2l @ c_2Ereal__topology_2Esequentially ) )
& ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Eprim__rec_2E_3C @ V3m @ V5n ) @ ( V0f @ V5n ) @ ( V1g @ V5n ) )
@ V2l
@ c_2Ereal__topology_2Esequentially )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0f @ V2l @ c_2Ereal__topology_2Esequentially ) )
& ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V6n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2E_3C_3D @ V6n @ V3m ) @ ( V0f @ V6n ) @ ( V1g @ V6n ) )
@ V2l
@ c_2Ereal__topology_2Esequentially )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V1g @ V2l @ c_2Ereal__topology_2Esequentially ) )
& ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V7n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Eprim__rec_2E_3C @ V7n @ V3m ) @ ( V0f @ V7n ) @ ( V1g @ V7n ) )
@ V2l
@ c_2Ereal__topology_2Esequentially )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V1g @ V2l @ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__CONG__WITHIN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1l: tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V4a: tyop_2Erealax_2Ereal] :
( ! [V5x: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V5x = V4a ) )
=> ( ( V3f @ V5x )
= ( V2g @ V5x ) ) )
=> ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( V3f @ V6x )
@ V1l
@ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V4a ) @ V0s ) )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V2g @ V1l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V4a ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__CONG__AT,axiom,
! [V0l: tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3a: tyop_2Erealax_2Ereal] :
( ! [V4x: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V4x = V3a ) )
=> ( ( V2f @ V4x )
= ( V1g @ V4x ) ) )
=> ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( V2f @ V5x )
@ V0l
@ ( c_2Ereal__topology_2Eat @ V3a ) )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V1g @ V0l @ ( c_2Ereal__topology_2Eat @ V3a ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__SEQUENTIAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1l @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
<=> ? [V2x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ! [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V2x @ V3n ) @ V0s )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V2x @ V1l @ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__CONTAINS__SEQUENTIAL__LIMIT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1x @ V3n ) @ V0s )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V1x @ V2l @ c_2Ereal__topology_2Esequentially ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2l @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__SEQUENTIAL__LIMITS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
<=> ! [V1x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal] :
( ( ! [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1x @ V3n ) @ V0s )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V1x @ V2l @ c_2Ereal__topology_2Esequentially ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2l @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__APPROACHABLE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V1s )
=> ( ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3y @ V0x ) ) @ V2e ) ) )
<=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EIN__CLOSURE__DELETE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EDELETE @ tyop_2Erealax_2Ereal @ V0s @ V1x ) ) )
= ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s ) ) ).
thf(thm_2Ereal__topology_2EDENSE__IMP__PERFECT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Eclosure @ V0s )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EDENSE__LIMIT__POINTS,axiom,
! [A_27a: $tType,V0s: tyop_2Erealax_2Ereal > $o,V1x: A_27a] :
( ( ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V0s ) ) )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
<=> ( ( c_2Ereal__topology_2Eclosure @ V0s )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ESEQ__OFFSET,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2k: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0f @ V1l @ c_2Ereal__topology_2Esequentially )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V3i: tyop_2Enum_2Enum] : ( V0f @ ( c_2Earithmetic_2E_2B @ V3i @ V2k ) )
@ V1l
@ c_2Ereal__topology_2Esequentially ) ) ).
thf(thm_2Ereal__topology_2ESEQ__OFFSET__NEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2k: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0f @ V1l @ c_2Ereal__topology_2Esequentially )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V3i: tyop_2Enum_2Enum] : ( V0f @ ( c_2Earithmetic_2E_2D @ V3i @ V2k ) )
@ V1l
@ c_2Ereal__topology_2Esequentially ) ) ).
thf(thm_2Ereal__topology_2ESEQ__OFFSET__REV,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2k: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V3i: tyop_2Enum_2Enum] : ( V0f @ ( c_2Earithmetic_2E_2B @ V3i @ V2k ) )
@ V1l
@ c_2Ereal__topology_2Esequentially )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0f @ V1l @ c_2Ereal__topology_2Esequentially ) ) ).
thf(thm_2Ereal__topology_2ESEQ__HARMONIC__OFFSET,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Erealax_2Einv @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ V0a ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ c_2Ereal__topology_2Esequentially ) ).
thf(thm_2Ereal__topology_2ESEQ__HARMONIC,axiom,
( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V0n: tyop_2Enum_2Enum] : ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ V0n ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ c_2Ereal__topology_2Esequentially ) ).
thf(thm_2Ereal__topology_2ECLOSED__CBALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EClosed @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) ) ).
thf(thm_2Ereal__topology_2EIN__INTERIOR__CBALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Einterior @ V1s ) )
<=> ? [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2e ) ) @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__BALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elimit__point__of @ V1y @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2e ) ) )
<=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__BALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) )
= ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__BALL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) )
= ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__CBALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) )
= ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__BALL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e )
=> ( ( c_2Ereal__topology_2Efrontier @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1e ) ) )
= ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1e ) ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__CBALL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Efrontier @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1e ) ) )
= ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1e ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__EQ__EMPTY,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( c_2Erealax_2Ereal__lt @ V1e @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__EMPTY,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V1e @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__EQ__SING,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
<=> ( V1e
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__SING,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( V1e
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPHERE__SING,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( V1e
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPHERE__EQ__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
<=> ( ( V2x = V0a )
& ( V1r
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__WITHIN__INTERIOR,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Einterior @ V1s ) )
=> ( ( c_2Ereal__topology_2Eeventually @ tyop_2Erealax_2Ereal @ V0p @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2x ) @ V1s ) )
= ( c_2Ereal__topology_2Eeventually @ tyop_2Erealax_2Ereal @ V0p @ ( c_2Ereal__topology_2Eat @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__WITHIN__INTERIOR,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Ereal__topology_2Einterior @ V2s ) )
=> ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3x ) @ V2s ) )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Eat @ V3x ) ) ) ) ).
thf(thm_2Ereal__topology_2ENETLIMIT__WITHIN__INTERIOR,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2Einterior @ V0s ) )
=> ( ( c_2Ereal__topology_2Enetlimit @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1x ) @ V0s ) )
= V1x ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__IMP__PERFECT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( (~)
@ ? [V2a: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) )
=> ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__IMP__PERFECT__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ? [V2a: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ( ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s )
= ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__EMPTY,axiom,
c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
=> ( c_2Ereal__topology_2Ebounded__def @ V0s ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__INTERIOR,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ( c_2Ereal__topology_2Ebounded__def @ ( c_2Ereal__topology_2Einterior @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ( c_2Ereal__topology_2Ebounded__def @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__CLOSURE__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
= ( c_2Ereal__topology_2Ebounded__def @ V0s ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__CBALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ebounded__def @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__BALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ebounded__def @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) ) ).
thf(thm_2Ereal__topology_2EFINITE__IMP__BOUNDED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
=> ( c_2Ereal__topology_2Ebounded__def @ V0s ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
<=> ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__BIGUNION,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V0f )
& ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0f )
=> ( c_2Ereal__topology_2Ebounded__def @ V1s ) ) )
=> ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__POS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
<=> ? [V1b: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1b )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V2x ) @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__POS__LT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
<=> ? [V1b: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1b )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V2x ) @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
| ( c_2Ereal__topology_2Ebounded__def @ V1t ) )
=> ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__DIFF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__INSERT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ V1s ) )
= ( c_2Ereal__topology_2Ebounded__def @ V1s ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__BIGINTER,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ? [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0f )
& ( c_2Ereal__topology_2Ebounded__def @ V1s ) )
=> ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2ENOT__BOUNDED__UNIV,axiom,
(~) @ ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ECOBOUNDED__IMP__UNBOUNDED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) )
=> ( (~) @ ( c_2Ereal__topology_2Ebounded__def @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V1s )
& ( c_2Ereal__topology_2Elinear @ V0f ) )
=> ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SCALING,axiom,
! [V0c: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V1s )
=> ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V0c @ V2x )
@ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__NEGATIONS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V1x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__TRANSLATION,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V1s )
=> ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V2x )
@ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__TRANSLATION__EQ,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V2x )
@ V1s ) )
= ( c_2Ereal__topology_2Ebounded__def @ V1s ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__DIFFS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t ) )
=> ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal_2Ereal__sub @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SUMS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t ) )
=> ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SUMS__IMAGE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2t: A_27a > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( V0f @ V3x ) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V2t ) ) ) )
& ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( V1g @ V4x ) @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V2t ) ) ) ) )
=> ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ ( V0f @ V5x ) @ ( V1g @ V5x ) ) @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V2t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SUMS__IMAGES,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > tyop_2Erealax_2Ereal,V1t: A_27a > $o,V2s: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V2s )
& ! [V3a: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3a @ V2s )
=> ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( V0f @ V4x @ V3a ) @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V1t ) ) ) ) ) )
=> ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Eiterate_2ESum @ A_27b @ V2s @ ( V0f @ V5x ) ) @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V1t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SUBSET__BALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ? [V2r: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2r )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2r ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SUBSET__CBALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ? [V2r: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2r )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2r ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EUNBOUNDED__INTER__COBOUNDED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( (~) @ ( c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V1t ) ) )
=> ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOBOUNDED__INTER__UNBOUNDED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) )
& ( (~) @ ( c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__BOUNDED__EQ__TRIVIAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esubspace @ V0s )
=> ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
<=> ( V0s
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__COMPONENTWISE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
= ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : V1x
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__HAS__SUP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V1x @ ( c_2Ereal_2Esup @ V0s ) ) )
& ! [V2b: tyop_2Erealax_2Ereal] :
( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V3x @ V2b ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esup @ V0s ) @ V2b ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUP__INSERT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V1s )
=> ( ( c_2Ereal_2Esup @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ V1s ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0x @ ( c_2Ereal_2Emax @ V0x @ ( c_2Ereal_2Esup @ V1s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__HAS__INF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2Einf @ V0s ) @ V1x ) )
& ! [V2b: tyop_2Erealax_2Ereal] :
( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V2b @ V3x ) )
=> ( c_2Ereal_2Ereal__lte @ V2b @ ( c_2Eiterate_2Einf @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINF__INSERT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V1s )
=> ( ( c_2Eiterate_2Einf @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ V1s ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0x @ ( c_2Ereal_2Emin @ V0x @ ( c_2Eiterate_2Einf @ V1s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSET__BALLS,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1a_27: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal,V3r_27: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2r ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a_27 @ V3r_27 ) ) )
<=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1a_27 ) ) @ V2r ) @ V3r_27 )
| ( c_2Ereal_2Ereal__lte @ V2r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
& ! [V4a: tyop_2Erealax_2Ereal,V5a_27: tyop_2Erealax_2Ereal,V6r: tyop_2Erealax_2Ereal,V7r_27: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V6r ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5a_27 @ V7r_27 ) ) )
<=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V5a_27 ) ) @ V6r ) @ V7r_27 )
| ( c_2Ereal_2Ereal__lte @ V6r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
& ! [V8a: tyop_2Erealax_2Ereal,V9a_27: tyop_2Erealax_2Ereal,V10r: tyop_2Erealax_2Ereal,V11r_27: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V10r ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V9a_27 @ V11r_27 ) ) )
<=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V9a_27 ) ) @ V10r ) @ V11r_27 )
| ( c_2Erealax_2Ereal__lt @ V10r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
& ! [V12a: tyop_2Erealax_2Ereal,V13a_27: tyop_2Erealax_2Ereal,V14r: tyop_2Erealax_2Ereal,V15r_27: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V12a @ V14r ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V13a_27 @ V15r_27 ) ) )
<=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V12a @ V13a_27 ) ) @ V14r ) @ V15r_27 )
| ( c_2Erealax_2Ereal__lt @ V14r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTER__BALLS__EQ__EMPTY,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2r ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V3s ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( c_2Ereal_2Ereal__lte @ V2r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( c_2Ereal_2Ereal__lte @ V3s @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ V2r @ V3s ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ) )
& ! [V4a: tyop_2Erealax_2Ereal,V5b: tyop_2Erealax_2Ereal,V6r: tyop_2Erealax_2Ereal,V7s: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V6r ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5b @ V7s ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( c_2Ereal_2Ereal__lte @ V6r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( c_2Erealax_2Ereal__lt @ V7s @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ V6r @ V7s ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V5b ) ) ) ) )
& ! [V8a: tyop_2Erealax_2Ereal,V9b: tyop_2Erealax_2Ereal,V10r: tyop_2Erealax_2Ereal,V11s: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V10r ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V9b @ V11s ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( c_2Erealax_2Ereal__lt @ V10r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( c_2Ereal_2Ereal__lte @ V11s @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ V10r @ V11s ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V9b ) ) ) ) )
& ! [V12a: tyop_2Erealax_2Ereal,V13b: tyop_2Erealax_2Ereal,V14r: tyop_2Erealax_2Ereal,V15s: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V12a @ V14r ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V13b @ V15s ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( c_2Erealax_2Ereal__lt @ V14r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( c_2Erealax_2Ereal__lt @ V15s @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V14r @ V15s ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V12a @ V13b ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__CLOSED__INTERVAL__CONVEX,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal,V4e: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4e )
& ( c_2Ereal_2Ereal__lte @ V4e @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V4e @ V2x ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V4e ) @ V3y ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__INTERVAL__MIDPOINT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Erealax_2Ereal__add @ V0a @ V1b ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecompact @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ENOT__INTERVAL__UNIV,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( (~)
@ ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( (~)
@ ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__INTERVAL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ebounded__def @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ebounded__def @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__CLOSED__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ebounded__def @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__INTERVAL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) )
= ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__CLOSED__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EClosed @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EOpen @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__INTERVAL__LEMMA,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V0a @ V2x )
& ( c_2Erealax_2Ereal__lt @ V2x @ V1b ) )
=> ? [V3d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3d )
& ! [V4x_27: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V4x_27 @ V2x ) ) @ V3d )
=> ( ( c_2Erealax_2Ereal__lt @ V0a @ V4x_27 )
& ( c_2Erealax_2Ereal__lt @ V4x_27 @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LT__MIN,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V2z @ ( c_2Ereal_2Emin @ V0x @ V1y ) )
<=> ( ( c_2Erealax_2Ereal__lt @ V2z @ V0x )
& ( c_2Erealax_2Ereal__lt @ V2z @ V1y ) ) ) ).
thf(thm_2Ereal__topology_2EINTERVAL__OPEN__SUBSET__CLOSED,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTER__INTERVAL,axiom,
! [V0d: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal,V3a: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1c @ V0d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Emax @ V3a @ V1c ) @ ( c_2Ereal_2Emin @ V2b @ V0d ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EENDS__IN__UNIT__INTERVAL,axiom,
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EENDS__IN__INTERVAL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ( (~)
@ ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3b @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ( (~)
@ ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
& ! [V4a: tyop_2Erealax_2Ereal,V5b: tyop_2Erealax_2Ereal] : ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4a @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V5b ) ) ) )
& ! [V6a: tyop_2Erealax_2Ereal,V7b: tyop_2Erealax_2Ereal] : ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7b @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6a @ V7b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDISJOINT__INTERVAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( c_2Erealax_2Ereal__lt @ V1b @ V0a )
| ( c_2Erealax_2Ereal__lt @ V3d @ V2c )
| ( c_2Erealax_2Ereal__lt @ V1b @ V2c )
| ( c_2Erealax_2Ereal__lt @ V3d @ V0a ) ) )
& ( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( c_2Erealax_2Ereal__lt @ V1b @ V0a )
| ( c_2Ereal_2Ereal__lte @ V3d @ V2c )
| ( c_2Ereal_2Ereal__lte @ V1b @ V2c )
| ( c_2Ereal_2Ereal__lte @ V3d @ V0a ) ) )
& ( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( c_2Ereal_2Ereal__lte @ V1b @ V0a )
| ( c_2Erealax_2Ereal__lt @ V3d @ V2c )
| ( c_2Ereal_2Ereal__lte @ V1b @ V2c )
| ( c_2Ereal_2Ereal__lte @ V3d @ V0a ) ) )
& ( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2c @ V3d ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( c_2Ereal_2Ereal__lte @ V1b @ V0a )
| ( c_2Ereal_2Ereal__lte @ V3d @ V2c )
| ( c_2Ereal_2Ereal__lte @ V1b @ V2c )
| ( c_2Ereal_2Ereal__lte @ V3d @ V0a ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSET__INTERVAL,axiom,
! [V0d: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal,V3a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1c @ V0d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ( ( c_2Ereal_2Ereal__lte @ V1c @ V0d )
=> ( ( c_2Ereal_2Ereal__lte @ V3a @ V1c )
& ( c_2Ereal_2Ereal__lte @ V0d @ V2b ) ) ) )
& ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1c @ V0d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V2b ) ) )
<=> ( ( c_2Ereal_2Ereal__lte @ V1c @ V0d )
=> ( ( c_2Erealax_2Ereal__lt @ V3a @ V1c )
& ( c_2Erealax_2Ereal__lt @ V0d @ V2b ) ) ) )
& ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1c @ V0d ) ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ( ( c_2Erealax_2Ereal__lt @ V1c @ V0d )
=> ( ( c_2Ereal_2Ereal__lte @ V3a @ V1c )
& ( c_2Ereal_2Ereal__lte @ V0d @ V2b ) ) ) )
& ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1c @ V0d ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V2b ) ) )
<=> ( ( c_2Erealax_2Ereal__lt @ V1c @ V0d )
=> ( ( c_2Ereal_2Ereal__lte @ V3a @ V1c )
& ( c_2Ereal_2Ereal__lte @ V0d @ V2b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERVAL__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V0a ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V0a ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSET__INTERVAL__IMP,axiom,
! [V0d: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal,V3a: tyop_2Erealax_2Ereal] :
( ( ( ( c_2Ereal_2Ereal__lte @ V3a @ V1c )
& ( c_2Ereal_2Ereal__lte @ V0d @ V2b ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1c @ V0d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) )
& ( ( ( c_2Erealax_2Ereal__lt @ V3a @ V1c )
& ( c_2Erealax_2Ereal__lt @ V0d @ V2b ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1c @ V0d ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V2b ) ) ) )
& ( ( ( c_2Ereal_2Ereal__lte @ V3a @ V1c )
& ( c_2Ereal_2Ereal__lte @ V0d @ V2b ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1c @ V0d ) ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) )
& ( ( ( c_2Ereal_2Ereal__lte @ V3a @ V1c )
& ( c_2Ereal_2Ereal__lte @ V0d @ V2b ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1c @ V0d ) ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V2b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERVAL__NE__EMPTY,axiom,
! [V0b: tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V0b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
<=> ( c_2Ereal_2Ereal__lte @ V1a @ V0b ) )
& ( ( (~)
@ ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V0b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
<=> ( c_2Erealax_2Ereal__lt @ V1a @ V0b ) ) ) ).
thf(thm_2Ereal__topology_2EINTERVAL__EQ__EMPTY,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V1b @ V0a )
<=> ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( ( c_2Ereal_2Ereal__lte @ V1b @ V0a )
<=> ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EREFLECT__INTERVAL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V2x )
@ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V1b ) @ ( c_2Erealax_2Ereal__neg @ V0a ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V3a: tyop_2Erealax_2Ereal,V4b: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V5x )
@ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V4b ) ) )
= ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V4b ) @ ( c_2Erealax_2Ereal__neg @ V3a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EIN__INTERVAL__REFLECT,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V2x ) @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V1b ) @ ( c_2Erealax_2Ereal__neg @ V0a ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
= ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) )
& ! [V3a: tyop_2Erealax_2Ereal,V4b: tyop_2Erealax_2Ereal,V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V5x ) @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V4b ) @ ( c_2Erealax_2Ereal__neg @ V3a ) ) ) )
= ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V4b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EIN__INTERVAL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V1b ) ) )
<=> ( ( c_2Erealax_2Ereal__lt @ V2a @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ V1b ) ) )
& ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ( ( c_2Ereal_2Ereal__lte @ V2a @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2Einterval,axiom,
! [V0b: tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EOPEN__interval @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V0b ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ V1a @ V2x ) @ ( c_2Erealax_2Ereal__lt @ V2x @ V0b ) ) ) ) )
& ( ( c_2Ereal__topology_2ECLOSED__interval @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V0b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V1a @ V3x ) @ ( c_2Ereal_2Ereal__lte @ V3x @ V0b ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__UNION__COMPACT__SUBSETS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ? [V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > $o] :
( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2Ecompact @ ( V1f @ V2n ) )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V1f @ V3n ) @ V0s )
& ! [V4n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V1f @ V4n ) @ ( c_2Ereal__topology_2Einterior @ ( V1f @ ( c_2Earithmetic_2E_2B @ V4n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ tyop_2Enum_2Enum
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( V1f @ V5n ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
= V0s )
& ! [V6k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V6k )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V6k @ V0s ) )
=> ? [V7N: tyop_2Enum_2Enum] :
! [V8n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V8n @ V7N )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V6k @ ( V1f @ V8n ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__UNION__COMPACT__SUBSETS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
=> ? [V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > $o] :
( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2Ecompact @ ( V1f @ V2n ) )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V1f @ V3n ) @ V0s )
& ! [V4n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V1f @ V4n ) @ ( V1f @ ( c_2Earithmetic_2E_2B @ V4n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ tyop_2Enum_2Enum
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( V1f @ V5n ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
= V0s )
& ! [V6k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V6k )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V6k @ V0s ) )
=> ? [V7N: tyop_2Enum_2Enum] :
! [V8n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V8n @ V7N )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V6k @ ( V1f @ V8n ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEPARATE__CLOSED__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ? [V2d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2d )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1t ) )
=> ( c_2Ereal_2Ereal__lte @ V2d @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x @ V4y ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEPARATE__COMPACT__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ? [V2d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2d )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1t ) )
=> ( c_2Ereal_2Ereal__lte @ V2d @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x @ V4y ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEPARATE__POINT__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V0s ) ) )
=> ? [V2d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2d )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V2d @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V3x ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ETRANSLATION__DIFF,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V3x )
@ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V1s @ V2t ) )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V4x )
@ V1s )
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V5x )
@ V2t ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__COMPACT__DIFFERENCES,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t ) )
=> ( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal_2Ereal__sub @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__CLOSED__DIFFERENCES,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t ) )
=> ( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Ereal_2Ereal__sub @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__SUMS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
| ( c_2Ereal__topology_2Ebounded__def @ V1t ) )
=> ( ( c_2Ereal__topology_2Eclosure
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V4x: tyop_2Erealax_2Ereal,V5y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V4x @ V5y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__COMPACT__SUMS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t ) )
=> ( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__CLOSED__SUMS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t ) )
=> ( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__NEGATIONS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
=> ( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V1x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__SCALING,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EClosed @ V0s )
=> ( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V1c @ V2x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ELEBESGUE__COVERING__LEMMA,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( (~)
@ ( V1c
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Erealax_2Ereal > $o ) ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V1c ) )
& ! [V2b: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2b @ V1c )
=> ( c_2Ereal__topology_2EOpen @ V2b ) ) )
=> ? [V3d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3d )
& ! [V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4t @ V0s )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Ediameter @ V4t ) @ V3d ) )
=> ? [V5b: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V5b @ V1c )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4t @ V5b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__SUMS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2Ebounded__def @ V1t ) )
=> ( c_2Ereal_2Ereal__lte
@ ( c_2Ereal__topology_2Ediameter
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) )
@ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2Ediameter @ V0s ) @ ( c_2Ereal__topology_2Ediameter @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__BALL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ediameter @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__lt @ V1r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V1r ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__CBALL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ediameter @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__lt @ V1r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V1r ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__LE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1d: tyop_2Erealax_2Ereal] :
( ( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
| ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1d ) )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V0s ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V2x @ V3y ) ) @ V1d ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Ediameter @ V0s ) @ V1d ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__EQ__0,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ( ( ( c_2Ereal__topology_2Ediameter @ V0s )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ? [V1a: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__SUBSET__CBALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ? [V1z: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1z @ ( c_2Ereal__topology_2Ediameter @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__SUBSET__CBALL__NONEMPTY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V1z: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1z @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1z @ ( c_2Ereal__topology_2Ediameter @ V0s ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ( ( c_2Ereal__topology_2Ediameter @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
= ( c_2Ereal__topology_2Ediameter @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ( c_2Ereal__topology_2Ebounded__def @ V1t ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2Ediameter @ V0s ) @ ( c_2Ereal__topology_2Ediameter @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__POS__LE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2Ediameter @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ediameter @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__EMPTY,axiom,
( ( c_2Ereal__topology_2Ediameter @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( V0f @ V2x ) )
= ( c_2Ereal_2Eabs @ V2x ) ) )
=> ( ( c_2Ereal__topology_2Ediameter @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
= ( c_2Ereal__topology_2Ediameter @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__BOUNDED__BOUND,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1x @ V2y ) ) @ ( c_2Ereal__topology_2Ediameter @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EDIAMETER__BOUNDED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ( ! [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1x @ V2y ) ) @ ( c_2Ereal__topology_2Ediameter @ V0s ) ) )
& ! [V3d: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3d )
& ( c_2Erealax_2Ereal__lt @ V3d @ ( c_2Ereal__topology_2Ediameter @ V0s ) ) )
=> ? [V4x: tyop_2Erealax_2Ereal,V5y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V0s )
& ( c_2Ereal_2Ereal__gt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V4x @ V5y ) ) @ V3d ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__AFFINITY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V1a @ ( c_2Erealax_2Ereal__mul @ V2c @ V3x ) )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__NEGATIONS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V1x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__SCALING,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V1c @ V2x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__NEGATIONS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
=> ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V1x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__SCALING,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected @ V0s )
=> ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V1c @ V2x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__INV,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
=> ( (~)
@ ( ( V0f @ V2x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ c_2Erealax_2Einv @ V0f ) @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__INV,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V1a ) )
& ( (~)
@ ( ( V0f @ V1a )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ c_2Erealax_2Einv @ V0f ) @ ( c_2Ereal__topology_2Eat @ V1a ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__WITHIN__INV,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2a ) @ V1s ) )
& ( (~)
@ ( ( V0f @ V2a )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ c_2Erealax_2Einv @ V0f ) @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2a ) @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__INV,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V1f @ V0net )
& ( (~)
@ ( ( V1f @ ( c_2Ereal__topology_2Enetlimit @ A_27a @ V0net ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ c_2Erealax_2Einv @ V1f ) @ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__INV,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net )
& ( (~)
@ ( V2l
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ c_2Erealax_2Einv @ V1f ) @ ( c_2Erealax_2Einv @ V2l ) @ V0net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__PRODUCT,axiom,
! [A_27a: $tType,V0f: tyop_2Erealax_2Ereal > A_27a > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2t )
& ! [V3i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3i @ V2t )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( V0f @ V4x @ V3i )
@ V1s ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Eproduct_2Eproduct @ A_27a @ V2t @ ( V0f @ V5x ) )
@ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__PRODUCT,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > A_27b > tyop_2Erealax_2Ereal,V2t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V2t )
& ! [V3i: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3i @ V2t )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V4x: A_27a] : ( V1f @ V4x @ V3i )
@ V0net ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V5x: A_27a] : ( c_2Eproduct_2Eproduct @ A_27b @ V2t @ ( V1f @ V5x ) )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__POW,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( V0f @ V3x )
@ V1s )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Epow @ ( V0f @ V4x ) @ V2n )
@ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__POW,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V3x: A_27a] : ( V1f @ V3x )
@ V0net )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ereal_2Epow @ ( V1f @ V4x ) @ V2n )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__MUL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V1c @ V0s )
& ( c_2Ereal__topology_2Econtinuous__on @ V2f @ V0s ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ ( V1c @ V3x ) @ ( V2f @ V3x ) )
@ V0s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__VMUL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2v: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V1c @ V0s )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ ( V1c @ V3x ) @ V2v )
@ V0s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__MUL,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2c: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V2c @ V0net )
& ( c_2Ereal__topology_2Econtinuous @ A_27a @ V1f @ V0net ) )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V2c @ V3x ) @ ( V1f @ V3x ) )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__VMUL,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1c: A_27a > tyop_2Erealax_2Ereal,V2v: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V1c @ V0net )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V1c @ V3x ) @ V2v )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__VMUL,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1c: A_27a > tyop_2Erealax_2Ereal,V2d: tyop_2Erealax_2Ereal,V3v: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1c @ V2d @ V0net )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V4x: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V1c @ V4x ) @ V3v )
@ ( c_2Erealax_2Ereal__mul @ V2d @ V3v )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__MUL,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3c: A_27a > tyop_2Erealax_2Ereal,V4d: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V3c @ V4d @ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V5x: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V3c @ V5x ) @ ( V1f @ V5x ) )
@ ( c_2Erealax_2Ereal__mul @ V4d @ V2l )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2EDISTANCE__ATTAINS__INF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ! [V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2x ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V3y ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDISTANCE__ATTAINS__SUP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ! [V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V3y ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2x ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ATTAINS__INF,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s ) )
=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
& ! [V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V2x ) @ ( V0f @ V3y ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ATTAINS__SUP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s ) )
=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
& ! [V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V3y ) @ ( V0f @ V2x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__ATTAINS__INF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
& ! [V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V1x @ V2y ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__ATTAINS__SUP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
& ! [V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V2y @ V1x ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__DIST,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( c_2Ereal__topology_2Econtinuous__on
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2x ) )
@ V1s ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__DIST,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2x ) )
@ ( c_2Ereal__topology_2Eat @ V1x ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__ABS,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ c_2Ereal_2Eabs @ ( c_2Ereal__topology_2Eat @ V0x ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__ABS__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs @ ( V0f @ V2x ) )
@ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ABS__COMPOSE,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V1f @ V0net )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V2x: A_27a] : ( c_2Ereal_2Eabs @ ( V1f @ V2x ) )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__RANGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
=> ! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4d )
& ! [V5x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x_27 @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V5x_27 @ V2x ) ) @ V4d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V0f @ V5x_27 ) @ ( V0f @ V2x ) ) ) @ V3e ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__RANGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V1x ) )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3d )
& ! [V4x_27: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V4x_27 @ V1x ) ) @ V3d )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V0f @ V4x_27 ) @ ( V0f @ V1x ) ) ) @ V2e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3x_27: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x_27 @ V0s )
& ( (~) @ ( V3x_27 = V1x ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V3x_27 @ V1x ) ) @ V2e ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ? [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ! [V3x_27: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V3x_27 @ V1x ) ) @ V2e )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x_27 @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__UNIFORM__LIMIT,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o] :
( ( ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V4n: A_27a] : ( c_2Ereal__topology_2Econtinuous__on @ ( V1f @ V4n ) @ V3s )
@ V0net )
& ! [V5e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5e )
=> ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V6n: A_27a] :
( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V3s ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V1f @ V6n @ V7x ) @ ( V2g @ V7x ) ) ) @ V5e ) ) )
@ V0net ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V2g @ V3s ) ) ).
thf(thm_2Ereal__topology_2EABS__TRIANGLE__LE,axiom,
! [V0e: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Eabs @ V1x ) @ ( c_2Ereal_2Eabs @ V2y ) ) @ V0e )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ V1x @ V2y ) ) @ V0e ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__UNIFORMLY__CONTINUOUS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2Ecompact @ V1s ) )
=> ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__UNIFORMLY__EQUICONTINUOUS,axiom,
! [V0fs: ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ! [V2x: tyop_2Erealax_2Ereal,V3e: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e ) )
=> ? [V4d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4d )
& ! [V5f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V6x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V5f @ V0fs )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x_27 @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6x_27 @ V2x ) ) @ V4d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V5f @ V6x_27 ) @ ( V5f @ V2x ) ) ) @ V3e ) ) ) )
& ( c_2Ereal__topology_2Ecompact @ V1s ) )
=> ! [V7e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V7e )
=> ? [V8d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V8d )
& ! [V9f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V10x: tyop_2Erealax_2Ereal,V11x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ V9f @ V0fs )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V11x_27 @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V11x_27 @ V10x ) ) @ V8d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V9f @ V11x_27 ) @ ( V9f @ V10x ) ) ) @ V7e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOPEN__IN__COMPONENTS,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ V1s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ V1s )
& ( c_2Ereal__topology_2Econnected @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ ( c_2Ereal__topology_2Ecomponents @ V0u ) ) ) ).
thf(thm_2Ereal__topology_2ECLOPEN__BIGUNION__COMPONENTS,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ V1s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ V1s ) )
=> ? [V2k: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( c_2Epred__set_2ESUBSET @ ( tyop_2Erealax_2Ereal > $o ) @ V2k @ ( c_2Ereal__topology_2Ecomponents @ V0u ) )
& ( V1s
= ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V2k ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__SUBSET__CLOPEN,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o,V2c: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ V1s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ V1s )
& ( c_2Ereal__topology_2Econnected @ V2c )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2c @ V0u )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2c @ V1s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2c @ V1s ) ) ).
thf(thm_2Ereal__topology_2EIN__COMPONENTS__BIGUNION__COMPLEMENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
=> ( ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V0s @ V1c )
= ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EDELETE @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2Ecomponents @ V0s ) @ V1c ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPONENTS__INTERMEDIATE__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ ( c_2Ereal__topology_2Ecomponents @ V2u ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V2u ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ ( c_2Ereal__topology_2Ecomponents @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EEXISTS__COMPONENT__SUPERSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Ereal__topology_2Econnected @ V1t ) )
=> ? [V2c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V2c ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPONENTS__UNIQUE__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1k: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Ereal__topology_2Ecomponents @ V0s )
= V1k )
<=> ( ( ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V1k )
= V0s )
& ! [V2c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2c @ V1k )
=> ( ( c_2Ereal__topology_2Econnected @ V2c )
& ( (~)
@ ( V2c
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V3c_27: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V3c_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2c @ V3c_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3c_27 @ V0s ) )
=> ( V3c_27 = V2c ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPONENTS__UNIQUE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1k: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V1k )
= V0s )
& ! [V2c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2c @ V1k )
=> ( ( c_2Ereal__topology_2Econnected @ V2c )
& ( (~)
@ ( V2c
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V3c_27: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V3c_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2c @ V3c_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3c_27 @ V0s ) )
=> ( V3c_27 = V2c ) ) ) ) )
=> ( ( c_2Ereal__topology_2Ecomponents @ V0s )
= V1k ) ) ).
thf(thm_2Ereal__topology_2ECOMPONENTS__MAXIMAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2c: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
& ( c_2Ereal__topology_2Econnected @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2c @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V2c ) ) ).
thf(thm_2Ereal__topology_2EIN__COMPONENTS__SELF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
<=> ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__EQ__COMPONENTS__SUBSET__SING__EXISTS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ? [V1a: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2Ecomponents @ V0s ) @ ( c_2Epred__set_2EINSERT @ ( tyop_2Erealax_2Ereal > $o ) @ V1a @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Erealax_2Ereal > $o ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__EQ__COMPONENTS__SUBSET__SING,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
= ( c_2Epred__set_2ESUBSET @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2Ecomponents @ V0s ) @ ( c_2Epred__set_2EINSERT @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Erealax_2Ereal > $o ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPONENTS__UNIV,axiom,
( ( c_2Ereal__topology_2Ecomponents @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EINSERT @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Erealax_2Ereal > $o ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPONENTS__EQ__SING__EXISTS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ? [V1a: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecomponents @ V0s )
= ( c_2Epred__set_2EINSERT @ ( tyop_2Erealax_2Ereal > $o ) @ V1a @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Erealax_2Ereal > $o ) ) ) )
<=> ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPONENTS__EQ__SING,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecomponents @ V0s )
= ( c_2Epred__set_2EINSERT @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Erealax_2Ereal > $o ) ) ) )
<=> ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPONENTS__EQ__SING__N__EXISTS,axiom,
( ! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecomponents @ V0s )
= ( c_2Epred__set_2EINSERT @ ( tyop_2Erealax_2Ereal > $o ) @ V0s @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Erealax_2Ereal > $o ) ) ) )
<=> ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ? [V2a: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecomponents @ V1s )
= ( c_2Epred__set_2EINSERT @ ( tyop_2Erealax_2Ereal > $o ) @ V2a @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Erealax_2Ereal > $o ) ) ) )
<=> ( ( c_2Ereal__topology_2Econnected @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__EQ__CONNECTED__COMPONENTS__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ! [V1c: tyop_2Erealax_2Ereal > $o,V2c_27: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
& ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2c_27 @ ( c_2Ereal__topology_2Ecomponents @ V0s ) ) )
=> ( V1c = V2c_27 ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPONENTS__EMPTY,axiom,
( ( c_2Ereal__topology_2Ecomponents @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Erealax_2Ereal > $o ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPONENTS__EQ__EMPTY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecomponents @ V0s )
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Erealax_2Ereal > $o ) ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPONENTS__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > $o,V2c_27: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
& ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2c_27 @ ( c_2Ereal__topology_2Ecomponents @ V0s ) ) )
=> ( ( V1c = V2c_27 )
<=> ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1c @ V2c_27 )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPONENTS__NONOVERLAP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > $o,V2c_27: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
& ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2c_27 @ ( c_2Ereal__topology_2Ecomponents @ V0s ) ) )
=> ( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1c @ V2c_27 )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( (~) @ ( V1c = V2c_27 ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENTS__FINITE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Ereal__topology_2Ecomponents @ V1s ) )
& ! [V2c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2c @ ( c_2Ereal__topology_2Ecomponents @ V1s ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2c ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENTS__GEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ! [V2c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2c @ ( c_2Ereal__topology_2Ecomponents @ V1s ) )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V2c )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2c ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__COMPONENTS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) ) )
=> ( c_2Ereal__topology_2Ecompact @ V1c ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__COMPONENTS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) ) )
=> ( c_2Ereal__topology_2EClosed @ V1c ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__COMPONENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1c ) ) ).
thf(thm_2Ereal__topology_2EJOINABLE__COMPONENTS__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2c1: tyop_2Erealax_2Ereal > $o,V3c2: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s )
& ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2c1 @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
& ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V3c2 @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2c1 @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V3c2 @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( V2c1 = V3c2 ) ) ).
thf(thm_2Ereal__topology_2EIN__COMPONENTS__MAXIMAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
<=> ( ( (~)
@ ( V1c
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1c @ V0s )
& ( c_2Ereal__topology_2Econnected @ V1c )
& ! [V2c_27: tyop_2Erealax_2Ereal > $o] :
( ( ( (~)
@ ( V2c_27
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1c @ V2c_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2c_27 @ V0s )
& ( c_2Ereal__topology_2Econnected @ V2c_27 ) )
=> ( V2c_27 = V1c ) ) ) ) ).
thf(thm_2Ereal__topology_2EIN__COMPONENTS__CONNECTED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
=> ( c_2Ereal__topology_2Econnected @ V1c ) ) ).
thf(thm_2Ereal__topology_2EIN__COMPONENTS__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1c @ V0s ) ) ).
thf(thm_2Ereal__topology_2EIN__COMPONENTS__NONEMPTY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1c @ ( c_2Ereal__topology_2Ecomponents @ V0s ) )
=> ( (~)
@ ( V1c
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EPAIRWISE__DISJOINT__COMPONENTS,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o] : ( c_2Ereal__topology_2Epairwise @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EDISJOINT @ tyop_2Erealax_2Ereal ) @ ( c_2Ereal__topology_2Ecomponents @ V0u ) ) ).
thf(thm_2Ereal__topology_2EBIGUNION__COMPONENTS,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o] :
( V0u
= ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecomponents @ V0u ) ) ) ).
thf(thm_2Ereal__topology_2EIN__COMPONENTS,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ ( c_2Ereal__topology_2Ecomponents @ V0u ) )
<=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0u )
& ( V1s
= ( c_2Ereal__topology_2Econnected__component @ V0u @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__INTERMEDIATE__SUBSET,axiom,
! [V0t: tyop_2Erealax_2Ereal > $o,V1u: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Econnected__component @ V1u @ V2a ) @ V0t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0t @ V1u ) )
=> ( ( c_2Ereal__topology_2Econnected__component @ V0t @ V2a )
= ( c_2Ereal__topology_2Econnected__component @ V1u @ V2a ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQUIVALENCE__RELATION,axiom,
! [V0R: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( V0R @ V2x @ V3y )
=> ( V0R @ V3y @ V2x ) )
& ! [V4x: tyop_2Erealax_2Ereal,V5y: tyop_2Erealax_2Ereal,V6z: tyop_2Erealax_2Ereal] :
( ( ( V0R @ V4x @ V5y )
& ( V0R @ V5y @ V6z ) )
=> ( V0R @ V4x @ V6z ) )
& ! [V7a: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7a @ V1s )
=> ? [V8t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V8t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7a @ V8t )
& ! [V9x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9x @ V8t )
=> ( V0R @ V7a @ V9x ) ) ) ) )
=> ! [V10a: tyop_2Erealax_2Ereal,V11b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component @ V1s @ V10a @ V11b )
=> ( V0R @ V10a @ V11b ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__CONNECTED__COMPONENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V2x ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) ) ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) ) ) ).
thf(thm_2Ereal__topology_2EBIGUNION__DIFF,axiom,
! [A_27a: $tType,V0s: ( A_27a > $o ) > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0s ) @ V1t )
= ( c_2Epred__set_2EBIGUNION @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V2x: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ ( c_2Epred__set_2EDIFF @ A_27a @ V2x @ V1t ) @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2x @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__CONNECTED__COMPONENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] : ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) ) ).
thf(thm_2Ereal__topology_2ECOMPLEMENT__CONNECTED__COMPONENT__BIGUNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) )
= ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EDELETE @ ( tyop_2Erealax_2Ereal > $o )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ tyop_2Erealax_2Ereal
@ ^ [V2y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V2y ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s ) ) )
@ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) ) ) ) ).
thf(thm_2Ereal__topology_2EBIGUNION__CONNECTED__COMPONENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) ) ) )
= V0s ) ).
thf(thm_2Ereal__topology_2EJOINABLE__CONNECTED__COMPONENT__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V2x ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V3y ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal__topology_2Econnected__component @ V0s @ V2x )
= ( c_2Ereal__topology_2Econnected__component @ V0s @ V3y ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__UNIQUE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1c )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1c @ V0s )
& ( c_2Ereal__topology_2Econnected @ V1c )
& ! [V3c_27: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V3c_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3c_27 @ V0s )
& ( c_2Ereal__topology_2Econnected @ V3c_27 ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3c_27 @ V1c ) ) )
=> ( ( c_2Ereal__topology_2Econnected__component @ V0s @ V2x )
= V1c ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__IDEMP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) @ V1x )
= ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__EQ__CONNECTED__COMPONENT__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s ) )
=> ( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x )
= ( c_2Ereal__topology_2Econnected__component @ V0s @ V2y ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x )
= ( c_2Ereal__topology_2Econnected__component @ V0s @ V2y ) )
<=> ( ( ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s ) ) )
| ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s )
& ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x @ V2y ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__SYM__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x @ V2y )
= ( c_2Ereal__topology_2Econnected__component @ V0s @ V2y @ V1x ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__OVERLAP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1a ) @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V2b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
<=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2b @ V0s )
& ( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1a )
= ( c_2Ereal__topology_2Econnected__component @ V0s @ V2b ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__NONOVERLAP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1a ) @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V2b ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V0s ) )
| ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2b @ V0s ) )
| ( (~)
@ ( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1a )
= ( c_2Ereal__topology_2Econnected__component @ V0s @ V2b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__DISJOINT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EDISJOINT @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1a ) @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V2b ) )
<=> ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V2b ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__CONNECTED__COMPONENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EClosed @ V0s )
=> ( c_2Ereal__topology_2EClosed @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) )
=> ( ( c_2Ereal__topology_2Econnected__component @ V0s @ V2y )
= ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EMPTY,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) @ V0x )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__EMPTY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__CONNECTED__COMPONENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__MONO,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V2x ) @ ( c_2Ereal__topology_2Econnected__component @ V1t @ V2x ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__MAXIMAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1t )
& ( c_2Ereal__topology_2Econnected @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V2x ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__IFF__CONNECTED__COMPONENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s ) )
=> ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x @ V2y ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__SELF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) )
=> ( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x )
= V0s ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__UNIV,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
<=> ( V0s
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__UNIV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0x )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__CONNECTED__COMPONENT__SET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x )
= V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x ) @ V0s ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__BIGUNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x )
= ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o )
@ ^ [V2t: tyop_2Erealax_2Ereal > $o] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ V2t @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal__topology_2Econnected @ V2t ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V2t ) @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2t @ V0s ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__SET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2y: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2y
@ ( c_2Ebool_2E_3F @ ( tyop_2Erealax_2Ereal > $o )
@ ^ [V3t: tyop_2Erealax_2Ereal > $o] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal__topology_2Econnected @ V3t ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3t @ V0s ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V3t ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V3t ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__OF__SUBSET,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3x: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1s @ V2t )
& ( c_2Ereal__topology_2Econnected__component @ V1s @ V3x @ V0y ) )
=> ( c_2Ereal__topology_2Econnected__component @ V2t @ V3x @ V0y ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__TRANS,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected__component @ V1s @ V2x @ V3y )
& ( c_2Ereal__topology_2Econnected__component @ V1s @ V3y @ V0z ) )
=> ( c_2Ereal__topology_2Econnected__component @ V1s @ V2x @ V0z ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__SYM,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x @ V2y )
=> ( c_2Ereal__topology_2Econnected__component @ V0s @ V2y @ V1x ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__REFL__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x @ V1x )
= ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__REFL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x @ V1x ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__COMPONENT__IN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component @ V0s @ V1x @ V2y )
=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EUPPER__LOWER__HEMICONTINUOUS__EXPLICIT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V2s )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V1t ) )
& ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V4u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V2s ) @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V4u ) ) ) ) ) )
& ! [V6u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V6u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V2s ) @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ V6u ) ) ) ) ) ) )
=> ! [V8x: tyop_2Erealax_2Ereal,V9e: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V2s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V9e )
& ( c_2Ereal__topology_2Ebounded__def @ ( V0f @ V8x ) )
& ( (~)
@ ( ( V0f @ V8x )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V10d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V10d )
& ! [V11x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V11x_27 @ V2s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8x @ V11x_27 ) ) @ V10d ) )
=> ( ! [V12y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V12y @ ( V0f @ V8x ) )
=> ? [V13y_27: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V13y_27 @ ( V0f @ V11x_27 ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V12y @ V13y_27 ) ) @ V9e ) ) )
& ! [V14y_27: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V14y_27 @ ( V0f @ V11x_27 ) )
=> ? [V15y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V15y @ ( V0f @ V8x ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V14y_27 @ V15y ) ) @ V9e ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__MAP__IFF__UPPER__HEMICONTINUOUS__PREIMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
=> ( ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3u ) ) )
<=> ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V4u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5y: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5y
@ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V2t )
@ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5y ) ) ) )
@ V4u ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__MAP__IFF__LOWER__HEMICONTINUOUS__PREIMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
=> ( ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3u ) ) )
<=> ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V4u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5y: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5y
@ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V2t )
@ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5y ) ) ) )
@ V4u ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOWER__HEMICONTINUOUS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2s: tyop_2Erealax_2Ereal > $o] :
( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V2s )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V1t ) )
=> ( ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V4u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V2s ) @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V4u ) ) ) ) ) )
<=> ! [V6u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V6u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V2s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ V6u ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EUPPER__HEMICONTINUOUS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2s: tyop_2Erealax_2Ereal > $o] :
( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V2s )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V1t ) )
=> ( ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V4u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V2s ) @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V4u ) ) ) ) ) )
<=> ! [V6u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V6u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V2s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ V6u ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__SUMS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
| ( c_2Ereal__topology_2EOpen @ V1t ) )
=> ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__TRANSLATION,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Einterior
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V2x )
@ V1s ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V3x )
@ ( c_2Ereal__topology_2Einterior @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__AFFINITY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
& ( (~)
@ ( V2c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V1a @ ( c_2Erealax_2Ereal__mul @ V2c @ V3x ) )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__TRANSLATION__EQ,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V2x )
@ V1s ) )
= ( c_2Ereal__topology_2EOpen @ V1s ) ) ).
thf(thm_2Ereal__topology_2EOPEN__TRANSLATION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V1a @ V2x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__NEGATIONS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V1x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__SCALING,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V1c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( c_2Ereal__topology_2EOpen @ V0s ) )
=> ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V1c @ V2x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected @ V1s )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2a ) ) ) ) )
& ? [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s )
& ( ( V0f @ V4x )
= V2a ) ) )
=> ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s )
=> ( ( V0f @ V5x )
= V2a ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN__IN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected @ V1s )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2a ) ) ) ) )
& ? [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s )
& ( ( V0f @ V4x )
= V2a ) ) )
=> ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s )
=> ( ( V0f @ V5x )
= V2a ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN__IN__CASES,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econnected @ V1s )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2a ) ) ) ) ) )
=> ( ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s )
=> ( (~)
@ ( ( V0f @ V4x )
= V2a ) ) )
| ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s )
=> ( ( V0f @ V5x )
= V2a ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN__AVOID,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2EOpen @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V2s )
& ( (~)
@ ( ( V0f @ V1x )
= V3a ) ) )
=> ? [V4e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4e )
& ! [V5y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V5y ) ) @ V4e )
=> ( (~)
@ ( ( V0f @ V5y )
= V3a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__AVOID,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V2s )
& ( (~)
@ ( ( V0f @ V1x )
= V3a ) ) )
=> ? [V4e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4e )
& ! [V5y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V2s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V5y ) ) @ V4e ) )
=> ( (~)
@ ( ( V0f @ V5y )
= V3a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__AVOID,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V1x ) )
& ( (~)
@ ( ( V0f @ V1x )
= V2a ) ) )
=> ? [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
& ! [V4y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V4y ) ) @ V3e )
=> ( (~)
@ ( ( V0f @ V4y )
= V2a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__AVOID,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1x ) @ V2s ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V2s )
& ( (~)
@ ( ( V0f @ V1x )
= V3a ) ) )
=> ? [V4e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4e )
& ! [V5y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V2s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V5y ) ) @ V4e ) )
=> ( (~)
@ ( ( V0f @ V5y )
= V3a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__IMAGE__SUBSET,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ! [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V2x ) )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V3x )
= ( V0f @ V4y ) )
=> ( V3x = V4y ) ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Einterior @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__TRANSLATION,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1z: tyop_2Erealax_2Ereal,V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V2f @ ( c_2Ereal__topology_2Eat @ ( c_2Erealax_2Ereal__add @ V0a @ V1z ) ) )
= ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( V2f @ ( c_2Erealax_2Ereal__add @ V0a @ V3x ) )
@ ( c_2Ereal__topology_2Eat @ V1z ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__COMPOSE__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V2g @ ( c_2Ereal__topology_2Eat @ V0x ) )
& ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V3h @ ( c_2Ereal__topology_2Eat @ ( V2g @ V0x ) ) )
& ! [V4y: tyop_2Erealax_2Ereal] :
( ( V2g @ ( V3h @ V4y ) )
= V4y )
& ( ( V3h @ ( V2g @ V0x ) )
= V0x ) )
=> ( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V1f @ ( c_2Ereal__topology_2Eat @ ( V2g @ V0x ) ) )
= ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( V1f @ ( V2g @ V5x ) )
@ ( c_2Ereal__topology_2Eat @ V0x ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__UNIFORMLY__CONTINUOUS__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2Ebounded__def @ V1s ) )
=> ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__IMP__CONTINUOUS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ! [V2x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecauchy @ V2x )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V2x @ V3n ) @ V1s ) )
=> ( c_2Ereal__topology_2Ecauchy @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2x ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__EXTENDS__TO__CLOSURE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V1s )
=> ? [V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Euniformly__continuous__on @ V2g @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
=> ( ( V2g @ V3x )
= ( V0f @ V3x ) ) )
& ! [V4h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V4h @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s )
=> ( ( V4h @ V5x )
= ( V0f @ V5x ) ) ) )
=> ! [V6x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
=> ( ( V4h @ V6x )
= ( V2g @ V6x ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__EXTENDS__TO__CLOSURE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ! [V2x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecauchy @ V2x )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V2x @ V3n ) @ V1s ) )
=> ( c_2Ereal__topology_2Ecauchy @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2x ) ) )
=> ? [V4g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V4g @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s )
=> ( ( V4g @ V5x )
= ( V0f @ V5x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__UNIQUENESS__LEMMA,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ! [V2x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecauchy @ V2x )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V2x @ V3n ) @ V1s ) )
=> ( c_2Ereal__topology_2Ecauchy @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2x ) ) )
=> ! [V4a: tyop_2Erealax_2Ereal,V5x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V6n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V5x @ V6n ) @ V1s )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V5x @ V4a @ c_2Ereal__topology_2Esequentially ) )
=> ? [V7l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5x ) @ V7l @ c_2Ereal__topology_2Esequentially )
& ! [V8y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V9n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V8y @ V9n ) @ V1s )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V8y @ V4a @ c_2Ereal__topology_2Esequentially ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V8y ) @ V7l @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IMP__CAUCHY__CONTINUOUS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2EClosed @ V1s ) )
=> ! [V2x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecauchy @ V2x )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V2x @ V3n ) @ V1s ) )
=> ( c_2Ereal__topology_2Ecauchy @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__IMP__CAUCHY__CONTINUOUS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V1s )
=> ! [V2x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecauchy @ V2x )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V2x @ V3n ) @ V1s ) )
=> ( c_2Ereal__topology_2Ecauchy @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CLOSURE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) )
=> ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__SEQUENTIALLY,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
<=> ! [V2x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3a @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ! [V4n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V2x @ V4n ) @ V1s )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V2x @ V3a @ c_2Ereal__topology_2Esequentially ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2x ) @ ( V0f @ V3a ) @ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
<=> ! [V2x: tyop_2Erealax_2Ereal,V3e: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e ) )
=> ? [V4d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4d )
& ! [V5y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5y @ V2x ) ) @ V4d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V5y ) @ ( V0f @ V2x ) ) ) @ V3e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE__CONSTANT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V2s )
& ( c_2Ereal__topology_2EClosed @ V2s ) )
=> ( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V2s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V1f @ V3x ) @ V0a ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__CONSTANT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2a ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AGREE__ON__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V1g @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
& ( c_2Ereal__topology_2Econtinuous__on @ V2h @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( ( V1g @ V3x )
= ( V2h @ V3x ) ) ) )
=> ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
=> ( ( V1g @ V4x )
= ( V2h @ V4x ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__CONSTANT__ON__CLOSURE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
=> ( ( V0f @ V3x )
= V2a ) ) )
=> ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
=> ( ( V0f @ V4x )
= V2a ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__GE__ON__CLOSURE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
=> ( c_2Ereal_2Ereal__lte @ V2a @ ( V0f @ V3x ) ) ) )
=> ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
=> ( c_2Ereal_2Ereal__lte @ V2a @ ( V0f @ V4x ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__LE__ON__CLOSURE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V3x ) @ V2a ) ) )
=> ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V4x ) @ V2a ) ) ) ).
thf(thm_2Ereal__topology_2EFORALL__IN__CLOSURE__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V2t )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) ) )
=> ( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2t ) )
<=> ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V2t ) ) ) ) ).
thf(thm_2Ereal__topology_2EFORALL__IN__CLOSURE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V2t )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2t ) ) )
=> ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__GT,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( (~)
@ ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__gt @ V1x @ V0a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__LT,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( (~)
@ ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Erealax_2Ereal__lt @ V1x @ V0a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__GE,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( (~)
@ ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__ge @ V1x @ V0a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__LE,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( (~)
@ ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__lte @ V1x @ V0a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__STANDARD__HYPERPLANE,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Einterior
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V1x @ V0a ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__HALFSPACE__GT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( V0a
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( V1b
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Ereal__topology_2Efrontier
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__gt @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0a @ V3x ) @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__HALFSPACE__LT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( V0a
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( V1b
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Ereal__topology_2Efrontier
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0a @ V3x ) @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__HALFSPACE__GE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( V0a
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( V1b
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Ereal__topology_2Efrontier
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__ge @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0a @ V3x ) @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__HALFSPACE__LE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( V0a
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( V1b
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Ereal__topology_2Efrontier
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0a @ V3x ) @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__HYPERPLANE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0a
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal__topology_2Einterior
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__HALFSPACE__COMPONENT__GT,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosure
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__gt @ V1x @ V0a ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__ge @ V2x @ V0a ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__HALFSPACE__COMPONENT__LT,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosure
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Erealax_2Ereal__lt @ V1x @ V0a ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__lte @ V2x @ V0a ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__HALFSPACE__GT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0a
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal__topology_2Eclosure
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__gt @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ereal_2Ereal__ge @ ( c_2Erealax_2Ereal__mul @ V0a @ V3x ) @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__HALFSPACE__LT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0a
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal__topology_2Eclosure
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V0a @ V3x ) @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__HALFSPACE__COMPONENT__GE,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Einterior
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__ge @ V1x @ V0a ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__gt @ V2x @ V0a ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__HALFSPACE__COMPONENT__LE,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Einterior
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__lte @ V1x @ V0a ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Erealax_2Ereal__lt @ V2x @ V0a ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__HALFSPACE__GE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0a
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal__topology_2Einterior
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__ge @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ereal_2Ereal__gt @ ( c_2Erealax_2Ereal__mul @ V0a @ V3x ) @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__HALFSPACE__LE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0a
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal__topology_2Einterior
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0a @ V3x ) @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__POSITIVE__ORTHANT,axiom,
( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V0x @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__INTERVAL__RIGHT,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Erealax_2Ereal__lt @ V0a @ V1x ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__INTERVAL__LEFT,axiom,
! [V0b: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Erealax_2Ereal__lt @ V1x @ V0b ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__POSITIVE__MULTIPLES,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V1c: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__mul @ V1c @ V2x ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1c ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__HALFSPACE__COMPONENT__GT,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__gt @ V1x @ V0a ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__HALFSPACE__GT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__gt @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__HALFSPACE__COMPONENT__LT,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Erealax_2Ereal__lt @ V1x @ V0a ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__HALFSPACE__LT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__HALFSPACE__COMPONENT__GE,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__ge @ V1x @ V0a ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__HALFSPACE__COMPONENT__LE,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__lte @ V1x @ V0a ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__STANDARD__HYPERPLANE,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V1x @ V0a ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__HYPERPLANE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosure
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0a @ V3x ) @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__HYPERPLANE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__HALFSPACE__GE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__ge @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__HALFSPACE__LE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V0a @ V2x ) @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__INTERVAL__RIGHT,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__lte @ V0a @ V1x ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__INTERVAL__LEFT,axiom,
! [V0b: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal_2Ereal__lte @ V1x @ V0b ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__LIFT__DOT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( c_2Ereal__topology_2Econtinuous__on
@ ^ [V2y: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V0a @ V2y )
@ V1s ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__LIFT__DOT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal
@ ^ [V2y: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V0a @ V2y )
@ ( c_2Ereal__topology_2Eat @ V1x ) ) ).
thf(thm_2Ereal__topology_2ELIM__LIFT__DOT,axiom,
! [V0net: tyop_2Ereal__topology_2Enet @ tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V2f @ V1l @ V0net )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal
@ ^ [V4y: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V3a @ ( V2f @ V4y ) )
@ ( c_2Erealax_2Ereal__mul @ V3a @ V1l )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2EPASTING__LEMMA__EXISTS__CLOSED,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1t: A_27a > tyop_2Erealax_2Ereal > $o,V2s: tyop_2Erealax_2Ereal > $o,V3k: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V3k )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2s
@ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ A_27a
@ ^ [V4i: A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( V1t @ V4i ) @ ( c_2Ebool_2EIN @ A_27a @ V4i @ V3k ) ) ) ) )
& ! [V5i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5i @ V3k )
=> ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ ( V1t @ V5i ) )
& ( c_2Ereal__topology_2Econtinuous__on @ ( V0f @ V5i ) @ ( V1t @ V5i ) ) ) )
& ! [V6i: A_27a,V7j: A_27a,V8x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ A_27a @ V6i @ V3k )
& ( c_2Ebool_2EIN @ A_27a @ V7j @ V3k )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2s @ ( V1t @ V6i ) ) @ ( V1t @ V7j ) ) ) )
=> ( ( V0f @ V6i @ V8x )
= ( V0f @ V7j @ V8x ) ) ) )
=> ? [V9g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V9g @ V2s )
& ! [V10x: tyop_2Erealax_2Ereal,V11i: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V11i @ V3k )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10x @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2s @ ( V1t @ V11i ) ) ) )
=> ( ( V9g @ V10x )
= ( V0f @ V11i @ V10x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EPASTING__LEMMA__CLOSED,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2t: A_27a > tyop_2Erealax_2Ereal > $o,V3s: tyop_2Erealax_2Ereal > $o,V4k: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V4k )
& ! [V5i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5i @ V4k )
=> ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3s ) @ ( V2t @ V5i ) )
& ( c_2Ereal__topology_2Econtinuous__on @ ( V0f @ V5i ) @ ( V2t @ V5i ) ) ) )
& ! [V6i: A_27a,V7j: A_27a,V8x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ A_27a @ V6i @ V4k )
& ( c_2Ebool_2EIN @ A_27a @ V7j @ V4k )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V3s @ ( V2t @ V6i ) ) @ ( V2t @ V7j ) ) ) )
=> ( ( V0f @ V6i @ V8x )
= ( V0f @ V7j @ V8x ) ) )
& ! [V9x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9x @ V3s )
=> ? [V10j: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V10j @ V4k )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9x @ ( V2t @ V10j ) )
& ( ( V1g @ V9x )
= ( V0f @ V10j @ V9x ) ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__OPEN,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o,V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V3s )
& ( c_2Ereal__topology_2EOpen @ V4t )
& ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V3s )
& ( c_2Ereal__topology_2Econtinuous__on @ V2g @ V4t )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V3s )
& ( (~) @ ( V0P @ V5x ) ) )
| ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V4t )
& ( V0P @ V5x ) ) )
=> ( ( V1f @ V5x )
= ( V2g @ V5x ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( V0P @ V6x ) @ ( V1f @ V6x ) @ ( V2g @ V6x ) )
@ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V3s @ V4t ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LOCAL__OPEN,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o,V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V3s @ V4t ) ) @ V3s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V3s @ V4t ) ) @ V4t )
& ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V3s )
& ( c_2Ereal__topology_2Econtinuous__on @ V2g @ V4t )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V3s )
& ( (~) @ ( V0P @ V5x ) ) )
| ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V4t )
& ( V0P @ V5x ) ) )
=> ( ( V1f @ V5x )
= ( V2g @ V5x ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( V0P @ V6x ) @ ( V1f @ V6x ) @ ( V2g @ V6x ) )
@ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V3s @ V4t ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__OPEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V1s )
& ( c_2Ereal__topology_2EOpen @ V2t )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2t ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V1s @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__LOCAL__OPEN,axiom,
! [V0t: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V2s @ V0t ) ) @ V2s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V2s @ V0t ) ) @ V0t )
& ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V2s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V0t ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V1f @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V2s @ V0t ) ) ) ).
thf(thm_2Ereal__topology_2EPASTING__LEMMA__EXISTS,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1t: A_27a > tyop_2Erealax_2Ereal > $o,V2s: tyop_2Erealax_2Ereal > $o,V3k: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2s
@ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ A_27a
@ ^ [V4i: A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( V1t @ V4i ) @ ( c_2Ebool_2EIN @ A_27a @ V4i @ V3k ) ) ) ) )
& ! [V5i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5i @ V3k )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ ( V1t @ V5i ) )
& ( c_2Ereal__topology_2Econtinuous__on @ ( V0f @ V5i ) @ ( V1t @ V5i ) ) ) )
& ! [V6i: A_27a,V7j: A_27a,V8x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ A_27a @ V6i @ V3k )
& ( c_2Ebool_2EIN @ A_27a @ V7j @ V3k )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2s @ ( V1t @ V6i ) ) @ ( V1t @ V7j ) ) ) )
=> ( ( V0f @ V6i @ V8x )
= ( V0f @ V7j @ V8x ) ) ) )
=> ? [V9g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V9g @ V2s )
& ! [V10x: tyop_2Erealax_2Ereal,V11i: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V11i @ V3k )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10x @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2s @ ( V1t @ V11i ) ) ) )
=> ( ( V9g @ V10x )
= ( V0f @ V11i @ V10x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EPASTING__LEMMA,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2t: A_27a > tyop_2Erealax_2Ereal > $o,V3s: tyop_2Erealax_2Ereal > $o,V4k: A_27a > $o] :
( ( ! [V5i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5i @ V4k )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3s ) @ ( V2t @ V5i ) )
& ( c_2Ereal__topology_2Econtinuous__on @ ( V0f @ V5i ) @ ( V2t @ V5i ) ) ) )
& ! [V6i: A_27a,V7j: A_27a,V8x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ A_27a @ V6i @ V4k )
& ( c_2Ebool_2EIN @ A_27a @ V7j @ V4k )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V3s @ ( V2t @ V6i ) ) @ ( V2t @ V7j ) ) ) )
=> ( ( V0f @ V6i @ V8x )
= ( V0f @ V7j @ V8x ) ) )
& ! [V9x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9x @ V3s )
=> ? [V10j: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V10j @ V4k )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9x @ ( V2t @ V10j ) )
& ( ( V1g @ V9x )
= ( V0f @ V10j @ V9x ) ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3s ) ) ).
thf(thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPOSITION__RIGHT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) @ V3t )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3t ) @ V4u )
& ! [V5k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5k @ V4u )
& ( c_2Ereal__topology_2Ecompact @ V5k ) )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f @ V6x ) @ V5k ) ) ) ) ) ) )
=> ! [V7k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V7k @ V3t )
& ( c_2Ereal__topology_2Ecompact @ V7k ) )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V8x ) @ V7k ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPOSITION__LEFT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s )
= V3t )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3t ) @ V4u )
& ! [V5k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5k @ V4u )
& ( c_2Ereal__topology_2Ecompact @ V5k ) )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f @ V6x ) @ V5k ) ) ) ) ) ) )
=> ! [V7k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V7k @ V4u )
& ( c_2Ereal__topology_2Ecompact @ V7k ) )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V3t ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1g @ V8x ) @ V7k ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EPROPER__MAP__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) @ V3t )
& ! [V5k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5k @ V3t )
& ( c_2Ereal__topology_2Ecompact @ V5k ) )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5k ) ) ) ) ) )
& ! [V7k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V7k @ V4u )
& ( c_2Ereal__topology_2Ecompact @ V7k ) )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V3t ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1g @ V8x ) @ V7k ) ) ) ) ) ) )
=> ! [V9k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V9k @ V4u )
& ( c_2Ereal__topology_2Ecompact @ V9k ) )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V10x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V10x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f @ V10x ) @ V9k ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPACT,axiom,
! [V0t: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V2s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1f @ V2s ) @ V0t )
& ( c_2Ereal__topology_2Ecompact @ V2s )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0t ) @ V3k ) )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1f @ V4x ) @ V3k ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__CONTINUOUS__IMAGE__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1s )
& ( ( V0f @ V2x )
= ( V0f @ V3y ) ) )
=> ( V2x = V3y ) )
=> ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V4t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4t @ V1s ) )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V4t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EPROPER__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
=> ( ! [V3k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3k @ V2t )
& ( c_2Ereal__topology_2Ecompact @ V3k ) )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3k ) ) ) ) ) )
<=> ( ! [V5k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V5k )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5k ) ) )
& ! [V6a: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6a @ V2t )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ V6a ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__CLOSED__MONOTONE__PREIMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V3c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3c )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3c ) ) )
& ! [V4y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V2t )
=> ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V4y ) ) ) ) ) ) )
=> ! [V6c: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V6c )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V6c @ V2t ) )
=> ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ V6c ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__OPEN__MONOTONE__PREIMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V3c: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3c )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3c ) ) )
& ! [V4y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V2t )
=> ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V4y ) ) ) ) ) ) )
=> ! [V6c: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V6c )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V6c @ V2t ) )
=> ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ V6c ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE__POINT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
=> ( ! [V3k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3k )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3k ) ) )
<=> ! [V4u: tyop_2Erealax_2Ereal > $o,V5y: tyop_2Erealax_2Ereal] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V4u )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V2t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5y ) ) ) )
@ V4u ) )
=> ? [V7v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V7v )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V7v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V8x ) @ V7v ) ) ) )
@ V4u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBIGUNION__GSPEC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,A_27h: $tType,A_27i: $tType] :
( ! [V0P: A_27a > $o,V1f: A_27a > A_27b > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27b
@ ( c_2Epred__set_2EGSPEC @ ( A_27b > $o ) @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ ( A_27b > $o ) @ $o @ ( V1f @ V2x ) @ ( V0P @ V2x ) ) ) )
= ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V3a: A_27b] :
( c_2Epair_2E_2C @ A_27b @ $o @ V3a
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V0P @ V4x ) @ ( c_2Ebool_2EIN @ A_27b @ V3a @ ( V1f @ V4x ) ) ) ) ) ) )
& ! [V5P: A_27c > A_27d > $o,V6f: A_27c > A_27d > A_27e > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27e
@ ( c_2Epred__set_2EGSPEC @ ( A_27e > $o ) @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d )
@ ( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ ( tyop_2Epair_2Eprod @ ( A_27e > $o ) @ $o )
@ ^ [V7x: A_27c,V8y: A_27d] : ( c_2Epair_2E_2C @ ( A_27e > $o ) @ $o @ ( V6f @ V7x @ V8y ) @ ( V5P @ V7x @ V8y ) ) ) ) )
= ( c_2Epred__set_2EGSPEC @ A_27e @ A_27e
@ ^ [V9a: A_27e] :
( c_2Epair_2E_2C @ A_27e @ $o @ V9a
@ ( c_2Ebool_2E_3F @ A_27c
@ ^ [V10x: A_27c] :
( c_2Ebool_2E_3F @ A_27d
@ ^ [V11y: A_27d] : ( c_2Ebool_2E_2F_5C @ ( V5P @ V10x @ V11y ) @ ( c_2Ebool_2EIN @ A_27e @ V9a @ ( V6f @ V10x @ V11y ) ) ) ) ) ) ) )
& ! [V12P: A_27f > A_27g > A_27h > $o,V13f: A_27f > A_27g > A_27h > A_27i > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27i
@ ( c_2Epred__set_2EGSPEC @ ( A_27i > $o ) @ ( tyop_2Epair_2Eprod @ A_27f @ ( tyop_2Epair_2Eprod @ A_27g @ A_27h ) )
@ ( c_2Epair_2EUNCURRY @ A_27f @ ( tyop_2Epair_2Eprod @ A_27g @ A_27h ) @ ( tyop_2Epair_2Eprod @ ( A_27i > $o ) @ $o )
@ ^ [V14x: A_27f] :
( c_2Epair_2EUNCURRY @ A_27g @ A_27h @ ( tyop_2Epair_2Eprod @ ( A_27i > $o ) @ $o )
@ ^ [V15y: A_27g,V16z: A_27h] : ( c_2Epair_2E_2C @ ( A_27i > $o ) @ $o @ ( V13f @ V14x @ V15y @ V16z ) @ ( V12P @ V14x @ V15y @ V16z ) ) ) ) ) )
= ( c_2Epred__set_2EGSPEC @ A_27i @ A_27i
@ ^ [V17a: A_27i] :
( c_2Epair_2E_2C @ A_27i @ $o @ V17a
@ ( c_2Ebool_2E_3F @ A_27f
@ ^ [V18x: A_27f] :
( c_2Ebool_2E_3F @ A_27g
@ ^ [V19y: A_27g] :
( c_2Ebool_2E_3F @ A_27h
@ ^ [V20z: A_27h] : ( c_2Ebool_2E_2F_5C @ ( V12P @ V18x @ V19y @ V20z ) @ ( c_2Ebool_2EIN @ A_27i @ V17a @ ( V13f @ V18x @ V19y @ V20z ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBIGINTER__GSPEC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,A_27h: $tType,A_27i: $tType] :
( ! [V0P: A_27a > $o,V1f: A_27a > A_27b > $o] :
( ( c_2Epred__set_2EBIGINTER @ A_27b
@ ( c_2Epred__set_2EGSPEC @ ( A_27b > $o ) @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ ( A_27b > $o ) @ $o @ ( V1f @ V2x ) @ ( V0P @ V2x ) ) ) )
= ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V3a: A_27b] :
( c_2Epair_2E_2C @ A_27b @ $o @ V3a
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V4x: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( V0P @ V4x ) @ ( c_2Ebool_2EIN @ A_27b @ V3a @ ( V1f @ V4x ) ) ) ) ) ) )
& ! [V5P: A_27c > A_27d > $o,V6f: A_27c > A_27d > A_27e > $o] :
( ( c_2Epred__set_2EBIGINTER @ A_27e
@ ( c_2Epred__set_2EGSPEC @ ( A_27e > $o ) @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d )
@ ( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ ( tyop_2Epair_2Eprod @ ( A_27e > $o ) @ $o )
@ ^ [V7x: A_27c,V8y: A_27d] : ( c_2Epair_2E_2C @ ( A_27e > $o ) @ $o @ ( V6f @ V7x @ V8y ) @ ( V5P @ V7x @ V8y ) ) ) ) )
= ( c_2Epred__set_2EGSPEC @ A_27e @ A_27e
@ ^ [V9a: A_27e] :
( c_2Epair_2E_2C @ A_27e @ $o @ V9a
@ ( c_2Ebool_2E_21 @ A_27c
@ ^ [V10x: A_27c] :
( c_2Ebool_2E_21 @ A_27d
@ ^ [V11y: A_27d] : ( c_2Emin_2E_3D_3D_3E @ ( V5P @ V10x @ V11y ) @ ( c_2Ebool_2EIN @ A_27e @ V9a @ ( V6f @ V10x @ V11y ) ) ) ) ) ) ) )
& ! [V12P: A_27f > A_27g > A_27h > $o,V13f: A_27f > A_27g > A_27h > A_27i > $o] :
( ( c_2Epred__set_2EBIGINTER @ A_27i
@ ( c_2Epred__set_2EGSPEC @ ( A_27i > $o ) @ ( tyop_2Epair_2Eprod @ A_27f @ ( tyop_2Epair_2Eprod @ A_27g @ A_27h ) )
@ ( c_2Epair_2EUNCURRY @ A_27f @ ( tyop_2Epair_2Eprod @ A_27g @ A_27h ) @ ( tyop_2Epair_2Eprod @ ( A_27i > $o ) @ $o )
@ ^ [V14x: A_27f] :
( c_2Epair_2EUNCURRY @ A_27g @ A_27h @ ( tyop_2Epair_2Eprod @ ( A_27i > $o ) @ $o )
@ ^ [V15y: A_27g,V16z: A_27h] : ( c_2Epair_2E_2C @ ( A_27i > $o ) @ $o @ ( V13f @ V14x @ V15y @ V16z ) @ ( V12P @ V14x @ V15y @ V16z ) ) ) ) ) )
= ( c_2Epred__set_2EGSPEC @ A_27i @ A_27i
@ ^ [V17a: A_27i] :
( c_2Epair_2E_2C @ A_27i @ $o @ V17a
@ ( c_2Ebool_2E_21 @ A_27f
@ ^ [V18x: A_27f] :
( c_2Ebool_2E_21 @ A_27g
@ ^ [V19y: A_27g] :
( c_2Ebool_2E_21 @ A_27h
@ ^ [V20z: A_27h] : ( c_2Emin_2E_3D_3D_3E @ ( V12P @ V18x @ V19y @ V20z ) @ ( c_2Ebool_2EIN @ A_27i @ V17a @ ( V13f @ V18x @ V19y @ V20z ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__FORALL__IN,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > tyop_2Erealax_2Ereal > $o] :
( ! [V2a: A_27a] :
( ( V0P @ V2a )
=> ( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( V1Q @ V2a @ V3x ) ) ) ) )
=> ( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V5a: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( V0P @ V5a ) @ ( V1Q @ V5a @ V4x ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__FORALL,axiom,
! [A_27a: $tType,V0Q: A_27a > tyop_2Erealax_2Ereal > $o] :
( ! [V1a: A_27a] :
( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( V0Q @ V1a @ V2x ) ) ) )
=> ( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V4a: A_27a] : ( V0Q @ V4a @ V3x ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
= ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__DIFF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t ) )
=> ( c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__DIFF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2EOpen @ V1t ) )
=> ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__BIGINTER,axiom,
! [V0s: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V0s )
& ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V0s )
=> ( c_2Ereal__topology_2EOpen @ V1t ) ) )
=> ( c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__BIGUNION,axiom,
! [V0s: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V0s )
& ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V0s )
=> ( c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EIN__BALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2e: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2e ) ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) @ V2e ) ) ).
thf(thm_2Ereal__topology_2EIN__CBALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2e: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2e ) ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) @ V2e ) ) ).
thf(thm_2Ereal__topology_2EIN__SPHERE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2e: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2e ) ) )
<=> ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) )
= V2e ) ) ).
thf(thm_2Ereal__topology_2EIN__BALL__0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e ) ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V0x ) @ V1e ) ) ).
thf(thm_2Ereal__topology_2EIN__CBALL__0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e ) ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V0x ) @ V1e ) ) ).
thf(thm_2Ereal__topology_2EIN__SPHERE__0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e ) ) )
<=> ( ( c_2Ereal_2Eabs @ V0x )
= V1e ) ) ).
thf(thm_2Ereal__topology_2EBALL__TRIVIAL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ECBALL__TRIVIAL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECENTRE__IN__CBALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e ) ) ).
thf(thm_2Ereal__topology_2EBALL__SUBSET__CBALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) ) ).
thf(thm_2Ereal__topology_2ESPHERE__SUBSET__CBALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSET__BALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1d: tyop_2Erealax_2Ereal,V2e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V1d @ V2e )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1d ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2e ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSET__CBALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1d: tyop_2Erealax_2Ereal,V2e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V1d @ V2e )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1d ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2e ) ) ) ) ).
thf(thm_2Ereal__topology_2EBALL__MAX__UNION,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal_2Emax @ V1r @ V2s ) ) )
= ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2s ) ) ) ) ).
thf(thm_2Ereal__topology_2EBALL__MIN__INTER,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal_2Emin @ V1r @ V2s ) ) )
= ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__MAX__UNION,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal_2Emax @ V1r @ V2s ) ) )
= ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__MIN__INTER,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1d: tyop_2Erealax_2Ereal,V2e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal_2Emin @ V1d @ V2e ) ) )
= ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1d ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2e ) ) ) ) ).
thf(thm_2Ereal__topology_2EBALL__TRANSLATION,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V0a @ V1x ) @ V2r ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3y: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V3y )
@ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2r ) ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__TRANSLATION,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V0a @ V1x ) @ V2r ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3y: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V3y )
@ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2r ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPHERE__TRANSLATION,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V0a @ V1x ) @ V2r ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3y: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V3y )
@ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2r ) ) ) ) ).
thf(thm_2Ereal__topology_2EBALL__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V3y: tyop_2Erealax_2Ereal] :
? [V4x: tyop_2Erealax_2Ereal] :
( ( V0f @ V4x )
= V3y )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( V0f @ V5x ) )
= ( c_2Ereal_2Eabs @ V5x ) ) )
=> ( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V1x ) @ V2r ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2r ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V3y: tyop_2Erealax_2Ereal] :
? [V4x: tyop_2Erealax_2Ereal] :
( ( V0f @ V4x )
= V3y )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( V0f @ V5x ) )
= ( c_2Ereal_2Eabs @ V5x ) ) )
=> ( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V1x ) @ V2r ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2r ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPHERE__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V3y: tyop_2Erealax_2Ereal] :
? [V4x: tyop_2Erealax_2Ereal] :
( ( V0f @ V4x )
= V3y )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( V0f @ V5x ) )
= ( c_2Ereal_2Eabs @ V5x ) ) )
=> ( ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V1x ) @ V2r ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2r ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBALL__SCALING,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0c )
=> ! [V1x: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0c @ V1x ) @ ( c_2Erealax_2Ereal__mul @ V0c @ V2r ) ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V0c @ V3x )
@ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2r ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__SCALING,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0c )
=> ! [V1x: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0c @ V1x ) @ ( c_2Erealax_2Ereal__mul @ V0c @ V2r ) ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V0c @ V3x )
@ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2r ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__DIFF__BALL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) )
= ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) ) ).
thf(thm_2Ereal__topology_2EBALL__UNION__SPHERE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) )
= ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) ) ).
thf(thm_2Ereal__topology_2ESPHERE__UNION__BALL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) )
= ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) ) ).
thf(thm_2Ereal__topology_2ECBALL__DIFF__SPHERE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) )
= ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__BALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EOpen @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) ) ).
thf(thm_2Ereal__topology_2ECENTRE__IN__BALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e ) ) ).
thf(thm_2Ereal__topology_2EOPEN__CONTAINS__BALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ? [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2e ) ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__CONTAINS__BALL__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
<=> ? [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2e ) ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EBALL__EQ__EMPTY,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( c_2Ereal_2Ereal__lte @ V1e @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2EBALL__EMPTY,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V1e @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__CONTAINS__CBALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ? [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2e ) ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__CONTAINS__CBALL__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
<=> ? [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2e ) ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPHERE__EQ__EMPTY,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( c_2Erealax_2Ereal__lt @ V1r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2ESPHERE__EMPTY,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V1r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
=> ( ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ENEGATIONS__BALL,axiom,
! [V0r: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V1x )
@ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0r ) ) )
= ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0r ) ) ) ).
thf(thm_2Ereal__topology_2ENEGATIONS__CBALL,axiom,
! [V0r: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V1x )
@ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0r ) ) )
= ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0r ) ) ) ).
thf(thm_2Ereal__topology_2ENEGATIONS__SPHERE,axiom,
! [V0r: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V1x )
@ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0r ) ) )
= ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0r ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__OPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1u ) @ V0s )
<=> ? [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V2t )
& ( V0s
= ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1u @ V2t ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__INTER__OPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V0s )
& ( c_2Ereal__topology_2EOpen @ V1t ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__OPEN__INTER,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V1s )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0u @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__OPEN__IN__TRANS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
& ( c_2Ereal__topology_2EOpen @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t ) ) ).
thf(thm_2Ereal__topology_2EOPEN__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ( c_2Ereal__topology_2EOpen @ V0s ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1u ) @ V0s )
<=> ? [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V2t )
& ( V0s
= ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1u @ V2t ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__SUBSET__EQ,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V1s )
=> ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ V1s )
= ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1s @ V0u ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__INTER__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t ) )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__CLOSED__INTER,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V1s )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0u @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ( c_2Ereal__topology_2EClosed @ V0s ) )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__SUBSET__TRANS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V2u ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__SUBSET__TRANS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V2u ) )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s ) ) ).
thf(thm_2Ereal__topology_2Eopen__in,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ V1s )
<=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1s @ V0u )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
=> ? [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
& ! [V4x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x_27 @ V0u )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4x_27 @ V2x ) ) @ V3e ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x_27 @ V1s ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__CONTAINS__BALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s )
<=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ? [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V3e ) ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__CONTAINS__CBALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s )
<=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ? [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V3e ) ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__TRANS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V1t ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V0s ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__TRANS__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ! [V2u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V2u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t ) )
<=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__OPEN__TRANS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s )
& ( c_2Ereal__topology_2EOpen @ V1t ) )
=> ( c_2Ereal__topology_2EOpen @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__TRANS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V1t ) )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__TRANS__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ! [V2u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V2u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t ) )
<=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__CLOSED__TRANS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t ) )
=> ( c_2Ereal__topology_2EClosed @ V0s ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__INTER__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1u: tyop_2Erealax_2Ereal > $o,V2v: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1u ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1u @ V0s ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2v @ V1u ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2v ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2v @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__OPEN__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t )
<=> ( ( c_2Ereal__topology_2EOpen @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__CLOSED__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
=> ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t )
<=> ( ( c_2Ereal__topology_2EClosed @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__SEGMENT__ALT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V0a = V1b ) )
=> ( ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2u: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2u ) @ V0a ) @ ( c_2Erealax_2Ereal__mul @ V2u @ V1b ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2u ) @ ( c_2Erealax_2Ereal__lt @ V2u @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Esegment,axiom,
! [V0b: tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V0b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2u: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2u ) @ V1a ) @ ( c_2Erealax_2Ereal__mul @ V2u @ V0b ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2u ) @ ( c_2Ereal_2Ereal__lte @ V2u @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V0b ) )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V0b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0b @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEGMENT__REFL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V0a ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V1a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V1a ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EIN__SEGMENT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
<=> ? [V3u: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3u )
& ( c_2Ereal_2Ereal__lte @ V3u @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( V2x
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V3u ) @ V0a ) @ ( c_2Erealax_2Ereal__mul @ V3u @ V1b ) ) ) ) )
& ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
<=> ( ( (~) @ ( V0a = V1b ) )
& ? [V4u: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4u )
& ( c_2Erealax_2Ereal__lt @ V4u @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( V2x
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V4u ) @ V0a ) @ ( c_2Erealax_2Ereal__mul @ V4u @ V1b ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEGMENT__SYM,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1b @ V0a ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) )
= ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3b @ V2a ) ) ) ) ).
thf(thm_2Ereal__topology_2EENDS__IN__SEGMENT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EENDS__NOT__IN__SEGMENT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEGMENT__CLOSED__OPEN,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEGMENT__OPEN__SUBSET__CLOSED,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EMIDPOINT__IN__SEGMENT,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Emidpoint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) )
<=> ( (~) @ ( V2a = V3b ) ) ) ) ).
thf(thm_2Ereal__topology_2EBETWEEN__IN__SEGMENT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebetween @ V0x @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) )
= ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__CONVEX__BOUND__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal,V3u: tyop_2Erealax_2Ereal,V4v: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0x @ V2a )
& ( c_2Ereal_2Ereal__lte @ V1y @ V2a )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3u )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4v )
& ( ( c_2Erealax_2Ereal__add @ V3u @ V4v )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V3u @ V0x ) @ ( c_2Erealax_2Ereal__mul @ V4v @ V1y ) ) @ V2a ) ) ).
thf(thm_2Ereal__topology_2EIN__SEGMENT__COMPONENT,axiom,
! [A_27a: $tType,V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3i: A_27a] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Emin @ V0a @ V1b ) @ V2x )
& ( c_2Ereal_2Ereal__lte @ V2x @ ( c_2Ereal_2Emax @ V0a @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEGMENT__TRANSLATION,axiom,
( ! [V0c: tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V0c @ V1a ) @ ( c_2Erealax_2Ereal__add @ V0c @ V2b ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0c @ V3x )
@ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) )
& ! [V4c: tyop_2Erealax_2Ereal,V5a: tyop_2Erealax_2Ereal,V6b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V4c @ V5a ) @ ( c_2Erealax_2Ereal__add @ V4c @ V6b ) ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V4c @ V7x )
@ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5a @ V6b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__SEGMENT__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V1a ) @ ( V0f @ V2b ) ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__SEGMENT__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V3x )
= ( V0f @ V4y ) )
=> ( V3x = V4y ) ) )
=> ( ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V1a ) @ ( V0f @ V2b ) ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EIN__OPEN__SEGMENT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
<=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( (~) @ ( V2x = V0a ) )
& ( (~) @ ( V2x = V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2EIN__OPEN__SEGMENT__ALT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
<=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( (~) @ ( V2x = V0a ) )
& ( (~) @ ( V2x = V1b ) )
& ( (~) @ ( V0a = V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__DIST__IN__CLOSED__SEGMENT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V0a ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V1b ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOLLINEAR__DIST__IN__OPEN__SEGMENT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V0a ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V1b ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__IN__OPEN__CLOSED__SEGMENT,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V0a ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V1b ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ) )
& ! [V3a: tyop_2Erealax_2Ereal,V4b: tyop_2Erealax_2Ereal,V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V4b ) ) )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x @ V3a ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V4b ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x @ V4b ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3a @ V4b ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__IN__CLOSED__SEGMENT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V0a ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V1b ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__IN__OPEN__SEGMENT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V0a ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V1b ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ( (~)
@ ? [V1e1: tyop_2Erealax_2Ereal > $o,V2e2: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V1e1 )
& ( c_2Ereal__topology_2EClosed @ V2e2 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 ) )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1e1 @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2e2 @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__OPEN__IN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ( (~)
@ ? [V1e1: tyop_2Erealax_2Ereal > $o,V2e2: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1e1 )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V2e2 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 ) )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( (~)
@ ( V1e1
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V2e2
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__OPEN__IN__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ( (~)
@ ? [V1e1: tyop_2Erealax_2Ereal > $o,V2e2: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1e1 )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V2e2 )
& ( ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 )
= V0s )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( (~)
@ ( V1e1
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V2e2
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__CLOSED__IN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ( (~)
@ ? [V1e1: tyop_2Erealax_2Ereal > $o,V2e2: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1e1 )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V2e2 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 ) )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( (~)
@ ( V1e1
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V2e2
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__CLOSED__IN__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ( (~)
@ ? [V1e1: tyop_2Erealax_2Ereal > $o,V2e2: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1e1 )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V2e2 )
& ( ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 )
= V0s )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( (~)
@ ( V1e1
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V2e2
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EEXISTS__DIFF,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o] :
( ? [V1s: A_27a > $o] : ( V0P @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V1s ) )
<=> ? [V2s: A_27a > $o] : ( V0P @ V2s ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__CLOPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t ) )
=> ( ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( V1t = V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__CLOSED__SET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
=> ( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ( (~)
@ ? [V1e1: tyop_2Erealax_2Ereal > $o,V2e2: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V1e1 )
& ( c_2Ereal__topology_2EClosed @ V2e2 )
& ( (~)
@ ( V1e1
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V2e2
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 )
= V0s )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__OPEN__SET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ( (~)
@ ? [V1e1: tyop_2Erealax_2Ereal > $o,V2e2: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V1e1 )
& ( c_2Ereal__topology_2EOpen @ V2e2 )
& ( (~)
@ ( V1e1
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V2e2
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 )
= V0s )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1e1 @ V2e2 )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__IFF__CONNECTABLE__POINTS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
<=> ! [V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2b @ V0s ) )
=> ? [V3t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3t @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V3t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2b @ V3t ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__EMPTY,axiom,
c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2ECONNECTED__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__REAL__LEMMA,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal,V3e1: tyop_2Erealax_2Ereal > $o,V4e2: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal_2Ereal__lte @ V1a @ V2b )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V1a ) @ V3e1 )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V2b ) @ V4e2 )
& ! [V5e: tyop_2Erealax_2Ereal,V6x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V1a @ V6x )
& ( c_2Ereal_2Ereal__lte @ V6x @ V2b )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5e ) )
=> ? [V7d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V7d )
& ! [V8y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V8y @ V6x ) ) @ V7d )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V8y ) @ ( V0f @ V6x ) ) ) @ V5e ) ) ) )
& ! [V9y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9y @ V3e1 )
=> ? [V10e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V10e )
& ! [V11y_27: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V11y_27 @ V9y ) ) @ V10e )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V11y_27 @ V3e1 ) ) ) )
& ! [V12y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V12y @ V4e2 )
=> ? [V13e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V13e )
& ! [V14y_27: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V14y_27 @ V12y ) ) @ V13e )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V14y_27 @ V4e2 ) ) ) )
& ( (~)
@ ? [V15x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V1a @ V15x )
& ( c_2Ereal_2Ereal__lte @ V15x @ V2b )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V15x ) @ V3e1 )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V15x ) @ V4e2 ) ) ) )
=> ? [V16x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V1a @ V16x )
& ( c_2Ereal_2Ereal__lte @ V16x @ V2b )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V16x ) @ V3e1 ) )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V16x ) @ V4e2 ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__SEGMENT,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected @ ( c_2Ereal__topology_2Eclosed__segment @ ( c_2Elist_2ECONS @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Elist_2ENIL @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected @ ( c_2Ereal__topology_2Eopen__segment @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__UNIV,axiom,
c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2ECLOPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2EOpen @ V0s ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
| ( V0s
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__BIGUNION,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0P )
=> ( c_2Ereal__topology_2Econnected @ V1s ) )
& ( (~)
@ ( ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0P )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0P ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( c_2Ereal__topology_2Econnected @ V1t )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__DIFF__OPEN__FROM__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V2u )
& ( c_2Ereal__topology_2EOpen @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t )
& ( c_2Ereal__topology_2Econnected @ V2u )
& ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V1t @ V0s ) ) )
=> ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V2u @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__DISJOINT__BIGUNION__OPEN__UNIQUE,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o,V1f_27: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Ereal__topology_2Epairwise @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EDISJOINT @ tyop_2Erealax_2Ereal ) @ V0f )
& ( c_2Ereal__topology_2Epairwise @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EDISJOINT @ tyop_2Erealax_2Ereal ) @ V1f_27 )
& ! [V2s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V0f )
=> ( ( c_2Ereal__topology_2EOpen @ V2s )
& ( c_2Ereal__topology_2Econnected @ V2s )
& ( (~)
@ ( V2s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V3s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V3s @ V1f_27 )
=> ( ( c_2Ereal__topology_2EOpen @ V3s )
& ( c_2Ereal__topology_2Econnected @ V3s )
& ( (~)
@ ( V3s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
& ( ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0f )
= ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V1f_27 ) ) )
=> ( V0f = V1f_27 ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__FROM__CLOSED__UNION__AND__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t )
& ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
& ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) )
=> ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( c_2Ereal__topology_2Econnected @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__FROM__OPEN__UNION__AND__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
& ( c_2Ereal__topology_2EOpen @ V1t )
& ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
& ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) )
=> ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( c_2Ereal__topology_2Econnected @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__INDUCTION,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1Q: tyop_2Erealax_2Ereal > $o,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V2s )
& ! [V3t: tyop_2Erealax_2Ereal > $o,V4a: tyop_2Erealax_2Ereal] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V3t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4a @ V3t ) )
=> ? [V5z: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5z @ V3t )
& ( V0P @ V5z ) ) )
& ! [V6a: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6a @ V2s )
=> ? [V7t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V7t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6a @ V7t )
& ! [V8x: tyop_2Erealax_2Ereal,V9y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V7t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9y @ V7t )
& ( V0P @ V8x )
& ( V0P @ V9y )
& ( V1Q @ V8x ) )
=> ( V1Q @ V9y ) ) ) ) )
=> ! [V10a: tyop_2Erealax_2Ereal,V11b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10a @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V11b @ V2s )
& ( V0P @ V10a )
& ( V0P @ V11b )
& ( V1Q @ V10a ) )
=> ( V1Q @ V11b ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__EQUIVALENCE__RELATION__GEN,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1R: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V2s )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( V1R @ V3x @ V4y )
=> ( V1R @ V4y @ V3x ) )
& ! [V5x: tyop_2Erealax_2Ereal,V6y: tyop_2Erealax_2Ereal,V7z: tyop_2Erealax_2Ereal] :
( ( ( V1R @ V5x @ V6y )
& ( V1R @ V6y @ V7z ) )
=> ( V1R @ V5x @ V7z ) )
& ! [V8t: tyop_2Erealax_2Ereal > $o,V9a: tyop_2Erealax_2Ereal] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V8t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9a @ V8t ) )
=> ? [V10z: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10z @ V8t )
& ( V0P @ V10z ) ) )
& ! [V11a: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V11a @ V2s )
=> ? [V12t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V12t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V11a @ V12t )
& ! [V13x: tyop_2Erealax_2Ereal,V14y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V13x @ V12t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V14y @ V12t )
& ( V0P @ V13x )
& ( V0P @ V14y ) )
=> ( V1R @ V13x @ V14y ) ) ) ) )
=> ! [V15a: tyop_2Erealax_2Ereal,V16b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V15a @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V16b @ V2s )
& ( V0P @ V15a )
& ( V0P @ V16b ) )
=> ( V1R @ V15a @ V16b ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__INDUCTION__SIMPLE,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V1s )
& ! [V2a: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2a @ V1s )
=> ? [V3t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2a @ V3t )
& ! [V4x: tyop_2Erealax_2Ereal,V5y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V3t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V3t )
& ( V0P @ V4x ) )
=> ( V0P @ V5y ) ) ) ) )
=> ! [V6a: tyop_2Erealax_2Ereal,V7b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6a @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7b @ V1s )
& ( V0P @ V6a ) )
=> ( V0P @ V7b ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__EQUIVALENCE__RELATION,axiom,
! [V0R: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V1s )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( V0R @ V2x @ V3y )
=> ( V0R @ V3y @ V2x ) )
& ! [V4x: tyop_2Erealax_2Ereal,V5y: tyop_2Erealax_2Ereal,V6z: tyop_2Erealax_2Ereal] :
( ( ( V0R @ V4x @ V5y )
& ( V0R @ V5y @ V6z ) )
=> ( V0R @ V4x @ V6z ) )
& ! [V7a: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7a @ V1s )
=> ? [V8t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V8t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7a @ V8t )
& ! [V9x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9x @ V8t )
=> ( V0R @ V7a @ V9x ) ) ) ) )
=> ! [V10a: tyop_2Erealax_2Ereal,V11b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10a @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V11b @ V1s ) )
=> ( V0R @ V10a @ V11b ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__SUBSET,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elimit__point__of @ V0x @ V1s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1s @ V2t ) )
=> ( c_2Ereal__topology_2Elimit__point__of @ V0x @ V2t ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__APPROACHABLE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elimit__point__of @ V0x @ V1s )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3x_27: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x_27 @ V1s )
& ( (~) @ ( V3x_27 = V0x ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x_27 @ V0x ) ) @ V2e ) ) ) ) ).
thf(thm_2Ereal__topology_2EAPPROACHABLE__LT__LE,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal] :
( ? [V2d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2d )
& ! [V3x: A_27a] :
( ( c_2Erealax_2Ereal__lt @ ( V1f @ V3x ) @ V2d )
=> ( V0P @ V3x ) ) )
<=> ? [V4d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4d )
& ! [V5x: A_27a] :
( ( c_2Ereal_2Ereal__lte @ ( V1f @ V5x ) @ V4d )
=> ( V0P @ V5x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__APPROACHABLE__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elimit__point__of @ V0x @ V1s )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3x_27: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x_27 @ V1s )
& ( (~) @ ( V3x_27 = V0x ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x_27 @ V0x ) ) @ V2e ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__CHOOSE__SIZE,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0c )
=> ? [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ V1x )
= V0c ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__UNIV,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Elimit__point__of @ V0x @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__LIMPT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__EMPTY,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( (~) @ ( c_2Ereal__topology_2Elimit__point__of @ V0x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ENO__LIMIT__POINT__IMP__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( (~)
@ ? [V1x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s ) )
=> ( c_2Ereal__topology_2EClosed @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__POSITIVE__ORTHANT,axiom,
( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V0x @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ) ) ).
thf(thm_2Ereal__topology_2EFINITE__SET__AVOID,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1s )
=> ? [V2d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2d )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
& ( (~) @ ( V3x = V0a ) ) )
=> ( c_2Ereal_2Ereal__lte @ V2d @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V3x ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIMIT__POINT__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
=> ( (~) @ ( c_2Ereal__topology_2Elimit__point__of @ V1a @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__SING,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( (~) @ ( c_2Ereal__topology_2Elimit__point__of @ V0x @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIMIT__POINT__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elimit__point__of @ V2x @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
<=> ( ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V0s )
| ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__INSERT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elimit__point__of @ V1x @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V2y @ V0s ) )
= ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__OF__LIMPTS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elimit__point__of @ V0x
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2y @ ( c_2Ereal__topology_2Elimit__point__of @ V2y @ V1s ) ) ) )
=> ( c_2Ereal__topology_2Elimit__point__of @ V0x @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__LIMPTS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EDISCRETE__IMP__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V0s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V3y @ V2x ) ) @ V1e ) )
=> ( V3y = V2x ) ) )
=> ( c_2Ereal__topology_2EClosed @ V0s ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__OF__UNIV,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Elimit__point__of @ V0x @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__OF__OPEN__IN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t )
& ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1t ) )
=> ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V1t ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__OF__OPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s ) )
=> ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__SING,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
<=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V0s )
& ( (~) @ ( c_2Ereal__topology_2Elimit__point__of @ V1a @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Einterior @ V0s )
= V0s )
<=> ( c_2Ereal__topology_2EOpen @ V0s ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__OPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( ( c_2Ereal__topology_2Einterior @ V0s )
= V0s ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__EMPTY,axiom,
( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__UNIV,axiom,
( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2EOPEN__INTERIOR,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Ereal__topology_2EOpen @ ( c_2Ereal__topology_2Einterior @ V0s ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__INTERIOR,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Einterior @ V0s ) )
= ( c_2Ereal__topology_2Einterior @ V0s ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Einterior @ V0s ) @ V0s ) ).
thf(thm_2Ereal__topology_2ESUBSET__INTERIOR__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Einterior @ V0s ) )
= ( c_2Ereal__topology_2EOpen @ V0s ) ) ).
thf(thm_2Ereal__topology_2ESUBSET__INTERIOR,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Einterior @ V0s ) @ ( c_2Ereal__topology_2Einterior @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__MAXIMAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s )
& ( c_2Ereal__topology_2EOpen @ V1t ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ ( c_2Ereal__topology_2Einterior @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__MAXIMAL__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Einterior @ V1t ) )
= ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__UNIQUE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s )
& ( c_2Ereal__topology_2EOpen @ V1t )
& ! [V2t_27: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2t_27 @ V0s )
& ( c_2Ereal__topology_2EOpen @ V2t_27 ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2t_27 @ V1t ) ) )
=> ( ( c_2Ereal__topology_2Einterior @ V0s )
= V1t ) ) ).
thf(thm_2Ereal__topology_2EIN__INTERIOR,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Einterior @ V1s ) )
<=> ? [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2e ) ) @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__SUBSET__INTERIOR,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Einterior @ V1t ) )
= ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
= ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Einterior @ V0s ) @ ( c_2Ereal__topology_2Einterior @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__FINITE__BIGINTER,axiom,
! [V0s: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V0s )
=> ( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0s ) )
= ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__BIGINTER__SUBSET,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0f ) ) @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ c_2Ereal__topology_2Einterior @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2EUNION__INTERIOR__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Einterior @ V0s ) @ ( c_2Ereal__topology_2Einterior @ V1t ) ) @ ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__EQ__EMPTY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Einterior @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s ) )
=> ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__EQ__EMPTY__ALT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Einterior @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V1t )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( (~)
@ ( ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V1t @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__LIMIT__POINT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2Einterior @ V0s ) )
=> ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__CLOSED__UNION__EMPTY__INTERIOR,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( ( c_2Ereal__topology_2Einterior @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
=> ( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
= ( c_2Ereal__topology_2Einterior @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__UNION__EQ__EMPTY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
| ( c_2Ereal__topology_2EClosed @ V1t ) )
=> ( ( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( ( c_2Ereal__topology_2Einterior @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( ( c_2Ereal__topology_2Einterior @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__UNIONS__OPEN__SUBSETS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o )
@ ^ [V1t: tyop_2Erealax_2Ereal > $o] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ V1t @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal__topology_2EOpen @ V1t ) @ ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s ) ) ) ) )
= ( c_2Ereal__topology_2Einterior @ V0s ) ) ).
thf(thm_2Ereal__topology_2EREAL__ARCH__INV,axiom,
! [V0e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0e )
<=> ? [V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( V1n = c_2Enum_2E0 ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ V1n ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ V1n ) ) @ V0e ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__POW__LBOUND,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ V0x ) ) @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) @ V1n ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__ARCH__POW,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x )
=> ? [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ V1y @ ( c_2Ereal_2Epow @ V0x @ V2n ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__ARCH__POW2,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
? [V1n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V1n ) ) ).
thf(thm_2Ereal__topology_2EREAL__ARCH__POW__INV,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y )
& ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ? [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Epow @ V0x @ V2n ) @ V1y ) ) ).
thf(thm_2Ereal__topology_2EFORALL__POS__MONO,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ! [V1d: tyop_2Erealax_2Ereal,V2e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V1d @ V2e )
& ( V0P @ V1d ) )
=> ( V0P @ V2e ) )
& ! [V3n: tyop_2Enum_2Enum] :
( ( (~) @ ( V3n = c_2Enum_2E0 ) )
=> ( V0P @ ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ V3n ) ) ) ) )
=> ! [V4e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4e )
=> ( V0P @ V4e ) ) ) ).
thf(thm_2Ereal__topology_2EFORALL__SUC,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ! [V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( V1n = c_2Enum_2E0 ) )
=> ( V0P @ V1n ) )
<=> ! [V2n: tyop_2Enum_2Enum] : ( V0P @ ( c_2Enum_2ESUC @ V2n ) ) ) ).
thf(thm_2Ereal__topology_2ELT__NZ,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
<=> ( (~) @ ( V0n = c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__ARCH__RDIV__EQ__0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1c )
& ! [V2m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2m )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2m ) @ V0x ) @ V1c ) ) )
=> ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__APPROACHABLE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3y @ V0x ) ) @ V2e ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__NONEMPTY__OPEN__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
<=> ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V2t )
& ( c_2Ereal__topology_2EOpen @ V2t ) )
=> ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V2t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__INTERIOR,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eclosure @ V0s )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Einterior @ V0s )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Ereal__topology_2EClosed @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__HULL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eclosure @ V0s )
= ( c_2Etopology_2Ehull @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2EClosed @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Eclosure @ V0s )
= V0s )
<=> ( c_2Ereal__topology_2EClosed @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
=> ( ( c_2Ereal__topology_2Eclosure @ V0s )
= V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
= ( c_2Ereal__topology_2Eclosure @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) ).
thf(thm_2Ereal__topology_2ESUBSET__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
= ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__INTER__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__BIGINTER__SUBSET,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0f ) ) @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o ) @ c_2Ereal__topology_2Eclosure @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__MINIMAL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ( c_2Ereal__topology_2EClosed @ V1t ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ V1t ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__MINIMAL__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V1t )
=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ V1t )
= ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__UNIQUE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ( c_2Ereal__topology_2EClosed @ V1t )
& ! [V2t_27: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V2t_27 )
& ( c_2Ereal__topology_2EClosed @ V2t_27 ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V2t_27 ) ) )
=> ( ( c_2Ereal__topology_2Eclosure @ V0s )
= V1t ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__EMPTY,axiom,
( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__UNIV,axiom,
( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__BIGUNION,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V0f )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0f ) )
= ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o )
@ ^ [V1s: tyop_2Erealax_2Ereal > $o] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( c_2Ereal__topology_2Eclosure @ V1s ) @ ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0f ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__EQ__EMPTY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Eclosure @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__SUBSET__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ V0s )
= ( c_2Ereal__topology_2EClosed @ V0s ) ) ).
thf(thm_2Ereal__topology_2EOPEN__INTER__CLOSURE__EQ__EMPTY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Eclosure @ V1t ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__OPEN__IN__INTER__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V2u ) )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) )
= ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__OPEN__INTER__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) )
= ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__INTER__CLOSURE__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__OPEN__INTER__SUPERSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
= ( c_2Ereal__topology_2Eclosure @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__COMPLEMENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ ( c_2Ereal__topology_2Einterior @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__COMPLEMENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__INTERMEDIATE__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) )
=> ( c_2Ereal__topology_2Econnected @ V1t ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V0s )
=> ( c_2Ereal__topology_2Econnected @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__UNION__STRONG,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( c_2Ereal__topology_2Econnected @ V1t )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__DIFF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Einterior @ V0s ) @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__OF__CLOSURE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elimit__point__of @ V0x @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
= ( c_2Ereal__topology_2Elimit__point__of @ V0x @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__LIMPT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s )
<=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1t ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__INTER__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t )
<=> ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Eclosure @ V1t ) )
= V1t ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__CLOSURE__IDEMP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) ) )
= ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__INTERIOR__IDEMP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ V0s ) ) ) )
= ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ENOWHERE__DENSE__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( ( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
& ( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ V1t ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ENOWHERE__DENSE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V1t )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V2u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V2u )
& ( (~)
@ ( V2u
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2u @ V1t )
& ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2u @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__CLOSURE__INTER__OPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
& ( c_2Ereal__topology_2EOpen @ V1t ) )
=> ( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) )
= ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__INTERIOR__UNION__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t ) )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) )
= ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ V0s ) ) @ ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ V1t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREGULAR__OPEN__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
= V0s )
& ( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ V1t ) )
= V1t ) )
=> ( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) )
= ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EREGULAR__CLOSED__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ V0s ) )
= V0s )
& ( ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ V1t ) )
= V1t ) )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) )
= ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EREGULAR__CLOSED__BIGUNION,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V0f )
& ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V0f )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ V1t ) )
= V1t ) ) )
=> ( ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0f ) ) )
= ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2EDIFF__CLOSURE__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EDENSE__OPEN__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V2u ) )
| ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2u ) @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V2u ) ) )
=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2u @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) )
<=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2u @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2u @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Ereal__topology_2EClosed @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__CLOSURES,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Efrontier @ V0s )
= ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__STRADDLE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal__topology_2Efrontier @ V1s ) )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ( ? [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V3x ) ) @ V2e ) )
& ? [V4x: tyop_2Erealax_2Ereal] :
( ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V4x ) ) @ V2e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__SUBSET__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ V0s ) @ V0s ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__EMPTY,axiom,
( ( c_2Ereal__topology_2Efrontier @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__UNIV,axiom,
( ( c_2Ereal__topology_2Efrontier @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__SUBSET__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ V0s ) @ V0s )
= ( c_2Ereal__topology_2EClosed @ V0s ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__COMPLEMENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Efrontier @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) )
= ( c_2Ereal__topology_2Efrontier @ V0s ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__DISJOINT__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ V0s ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
<=> ( c_2Ereal__topology_2EOpen @ V0s ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__INTER__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ V0s ) @ ( c_2Ereal__topology_2Efrontier @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__UNION__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ V0s ) @ ( c_2Ereal__topology_2Efrontier @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__INTERIORS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Efrontier @ V0s )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ ( c_2Ereal__topology_2Einterior @ V0s ) ) @ ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__FRONTIER__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__FRONTIER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Efrontier @ V0s ) )
= ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( c_2Ereal__topology_2Eclosure @ ( c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__FRONTIER__EMPTY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
| ( c_2Ereal__topology_2EClosed @ V0s ) )
=> ( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Efrontier @ V0s ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__FRONTIER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
| ( c_2Ereal__topology_2EClosed @ V0s ) )
=> ( ( c_2Ereal__topology_2Efrontier @ ( c_2Ereal__topology_2Efrontier @ V0s ) )
= ( c_2Ereal__topology_2Efrontier @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__FRONTIER__FRONTIER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Efrontier @ ( c_2Ereal__topology_2Efrontier @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) )
= ( c_2Ereal__topology_2Efrontier @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EUNION__FRONTIER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ V0s ) @ ( c_2Ereal__topology_2Efrontier @ V1t ) )
= ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) @ ( c_2Ereal__topology_2Efrontier @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ V0s ) @ ( c_2Ereal__topology_2Efrontier @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__INTER__FRONTIER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V0s )
& ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Efrontier @ V1t ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTERIOR__CLOSED__EQ__EMPTY__AS__FRONTIER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( ( c_2Ereal__topology_2Einterior @ V0s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
<=> ? [V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V1t )
& ( V0s
= ( c_2Ereal__topology_2Efrontier @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ ( c_2Ereal__topology_2Eclosure @ V1t ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
=> ( ( c_2Ereal__topology_2Efrontier @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
= ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ V0s ) @ ( c_2Ereal__topology_2Efrontier @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__UNION__FRONTIER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eclosure @ V0s )
= ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__INTERIOR__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ ( c_2Ereal__topology_2Einterior @ V0s ) ) @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__CLOSURE__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) ).
thf(thm_2Ereal__topology_2ESET__DIFF__FRONTIER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Efrontier @ V0s ) )
= ( c_2Ereal__topology_2Einterior @ V0s ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__INTER__SUBSET__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V0s ) @ ( c_2Ereal__topology_2Efrontier @ V1t ) ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ V0s ) @ ( c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2Enet__tybij,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( c_2Ereal__topology_2Emk__net @ A_27a @ ( c_2Ereal__topology_2Enetord @ A_27a @ V0a ) )
= V0a )
& ! [V1r: A_27b > A_27b > $o] :
( ! [V2x: A_27b,V3y: A_27b] :
( ! [V4z: A_27b] :
( ( V1r @ V4z @ V2x )
=> ( V1r @ V4z @ V3y ) )
| ! [V5z: A_27b] :
( ( V1r @ V5z @ V3y )
=> ( V1r @ V5z @ V2x ) ) )
<=> ( ( c_2Ereal__topology_2Enetord @ A_27b @ ( c_2Ereal__topology_2Emk__net @ A_27b @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Ereal__topology_2ENET,axiom,
! [A_27a: $tType,V0n: tyop_2Ereal__topology_2Enet @ A_27a,V1x: A_27a,V2y: A_27a] :
( ! [V3z: A_27a] :
( ( c_2Ereal__topology_2Enetord @ A_27a @ V0n @ V3z @ V1x )
=> ( c_2Ereal__topology_2Enetord @ A_27a @ V0n @ V3z @ V2y ) )
| ! [V4z: A_27a] :
( ( c_2Ereal__topology_2Enetord @ A_27a @ V0n @ V4z @ V2y )
=> ( c_2Ereal__topology_2Enetord @ A_27a @ V0n @ V4z @ V1x ) ) ) ).
thf(thm_2Ereal__topology_2EOLDNET,axiom,
! [A_27a: $tType,V0n: tyop_2Ereal__topology_2Enet @ A_27a,V1x: A_27a,V2y: A_27a] :
( ( ( c_2Ereal__topology_2Enetord @ A_27a @ V0n @ V1x @ V1x )
& ( c_2Ereal__topology_2Enetord @ A_27a @ V0n @ V2y @ V2y ) )
=> ? [V3z: A_27a] :
( ( c_2Ereal__topology_2Enetord @ A_27a @ V0n @ V3z @ V3z )
& ! [V4w: A_27a] :
( ( c_2Ereal__topology_2Enetord @ A_27a @ V0n @ V4w @ V3z )
=> ( ( c_2Ereal__topology_2Enetord @ A_27a @ V0n @ V4w @ V1x )
& ( c_2Ereal__topology_2Enetord @ A_27a @ V0n @ V4w @ V2y ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ENET__DILEMMA,axiom,
! [A_27a: $tType,V0Q: A_27a > $o,V1P: A_27a > $o,V2net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( ? [V3a: A_27a] :
( ? [V4x: A_27a] : ( c_2Ereal__topology_2Enetord @ A_27a @ V2net @ V4x @ V3a )
& ! [V5x: A_27a] :
( ( c_2Ereal__topology_2Enetord @ A_27a @ V2net @ V5x @ V3a )
=> ( V1P @ V5x ) ) )
& ? [V6b: A_27a] :
( ? [V7x: A_27a] : ( c_2Ereal__topology_2Enetord @ A_27a @ V2net @ V7x @ V6b )
& ! [V8x: A_27a] :
( ( c_2Ereal__topology_2Enetord @ A_27a @ V2net @ V8x @ V6b )
=> ( V0Q @ V8x ) ) ) )
=> ? [V9c: A_27a] :
( ? [V10x: A_27a] : ( c_2Ereal__topology_2Enetord @ A_27a @ V2net @ V10x @ V9c )
& ! [V11x: A_27a] :
( ( c_2Ereal__topology_2Enetord @ A_27a @ V2net @ V11x @ V9c )
=> ( ( V1P @ V11x )
& ( V0Q @ V11x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EAT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Enetord @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0a ) @ V1x @ V2y )
<=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V0a ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V0a ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2y @ V0a ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EAT__INFINITY,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Enetord @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eat__infinity @ V0x @ V1y )
= ( c_2Ereal_2Ereal__ge @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) ) ).
thf(thm_2Ereal__topology_2EAT__POSINFINITY,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Enetord @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eat__posinfinity @ V0x @ V1y )
= ( c_2Ereal_2Ereal__ge @ V0x @ V1y ) ) ).
thf(thm_2Ereal__topology_2EAT__NEGINFINITY,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Enetord @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eat__neginfinity @ V0x @ V1y )
= ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ).
thf(thm_2Ereal__topology_2ESEQUENTIALLY,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Enetord @ tyop_2Enum_2Enum @ c_2Ereal__topology_2Esequentially @ V0m @ V1n )
= ( c_2Earithmetic_2E_3E_3D @ V0m @ V1n ) ) ).
thf(thm_2Ereal__topology_2EWITHIN,axiom,
! [A_27a: $tType,V0n: tyop_2Ereal__topology_2Enet @ A_27a,V1s: A_27a > $o,V2x: A_27a,V3y: A_27a] :
( ( c_2Ereal__topology_2Enetord @ A_27a @ ( c_2Ereal__topology_2Ewithin @ A_27a @ V0n @ V1s ) @ V2x @ V3y )
<=> ( ( c_2Ereal__topology_2Enetord @ A_27a @ V0n @ V2x @ V3y )
& ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EIN__DIRECTION,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1v: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Enetord @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ein__direction @ V0a @ V1v ) @ V2x @ V3y )
<=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V0a ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V0a ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3y @ V0a ) ) )
& ? [V4c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4c )
& ( ( c_2Ereal_2Ereal__sub @ V2x @ V0a )
= ( c_2Erealax_2Ereal__mul @ V4c @ V1v ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EWITHIN__UNIV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0x ) @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
= ( c_2Ereal__topology_2Eat @ V0x ) ) ).
thf(thm_2Ereal__topology_2EWITHIN__WITHIN,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Ereal__topology_2Ewithin @ A_27a @ ( c_2Ereal__topology_2Ewithin @ A_27a @ V0net @ V1s ) @ V2t )
= ( c_2Ereal__topology_2Ewithin @ A_27a @ V0net @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2ETRIVIAL__LIMIT__WITHIN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Etrivial__limit @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1a ) @ V0s ) )
<=> ( (~) @ ( c_2Ereal__topology_2Elimit__point__of @ V1a @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT,axiom,
! [V0a: tyop_2Erealax_2Ereal] : ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0a ) ) ) ).
thf(thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__INFINITY,axiom,
(~) @ ( c_2Ereal__topology_2Etrivial__limit @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eat__infinity ) ).
thf(thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__POSINFINITY,axiom,
(~) @ ( c_2Ereal__topology_2Etrivial__limit @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eat__posinfinity ) ).
thf(thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__NEGINFINITY,axiom,
(~) @ ( c_2Ereal__topology_2Etrivial__limit @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eat__neginfinity ) ).
thf(thm_2Ereal__topology_2ETRIVIAL__LIMIT__SEQUENTIALLY,axiom,
(~) @ ( c_2Ereal__topology_2Etrivial__limit @ tyop_2Enum_2Enum @ c_2Ereal__topology_2Esequentially ) ).
thf(thm_2Ereal__topology_2ELIM__WITHIN__CLOSED__TRIVIAL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V1s )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ V1s ) ) )
=> ( c_2Ereal__topology_2Etrivial__limit @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0a ) @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ENONTRIVIAL__LIMIT__WITHIN,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1s: A_27a > $o] :
( ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net )
=> ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ ( c_2Ereal__topology_2Ewithin @ A_27a @ V0net @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__HAPPENS,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1p: A_27a > $o] :
( ( c_2Ereal__topology_2Eeventually @ A_27a @ V1p @ V0net )
=> ( ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net )
| ? [V2x: A_27a] : ( V1p @ V2x ) ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__WITHIN__LE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2p: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eeventually @ tyop_2Erealax_2Ereal @ V2p @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1a ) @ V0s ) )
<=> ? [V3d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3d )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4x @ V1a ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4x @ V1a ) ) @ V3d ) )
=> ( V2p @ V4x ) ) ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__WITHIN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2p: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eeventually @ tyop_2Erealax_2Ereal @ V2p @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1a ) @ V0s ) )
<=> ? [V3d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3d )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4x @ V1a ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4x @ V1a ) ) @ V3d ) )
=> ( V2p @ V4x ) ) ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__AT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1p: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eeventually @ tyop_2Erealax_2Ereal @ V1p @ ( c_2Ereal__topology_2Eat @ V0a ) )
<=> ? [V2d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2d )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x @ V0a ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x @ V0a ) ) @ V2d ) )
=> ( V1p @ V3x ) ) ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__SEQUENTIALLY,axiom,
! [V0p: tyop_2Enum_2Enum > $o] :
( ( c_2Ereal__topology_2Eeventually @ tyop_2Enum_2Enum @ V0p @ c_2Ereal__topology_2Esequentially )
<=> ? [V1N: tyop_2Enum_2Enum] :
! [V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1N @ V2n )
=> ( V0p @ V2n ) ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__AT__INFINITY,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eeventually @ tyop_2Erealax_2Ereal @ V0p @ c_2Ereal__topology_2Eat__infinity )
<=> ? [V1b: tyop_2Erealax_2Ereal] :
! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__ge @ ( c_2Ereal_2Eabs @ V2x ) @ V1b )
=> ( V0p @ V2x ) ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__AT__POSINFINITY,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eeventually @ tyop_2Erealax_2Ereal @ V0p @ c_2Ereal__topology_2Eat__posinfinity )
<=> ? [V1b: tyop_2Erealax_2Ereal] :
! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__ge @ V2x @ V1b )
=> ( V0p @ V2x ) ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__AT__NEGINFINITY,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eeventually @ tyop_2Erealax_2Ereal @ V0p @ c_2Ereal__topology_2Eat__neginfinity )
<=> ? [V1b: tyop_2Erealax_2Ereal] :
! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V2x @ V1b )
=> ( V0p @ V2x ) ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__AT__INFINITY__POS,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eeventually @ tyop_2Erealax_2Ereal @ V0p @ c_2Ereal__topology_2Eat__infinity )
<=> ? [V1b: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1b )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__ge @ ( c_2Ereal_2Eabs @ V2x ) @ V1b )
=> ( V0p @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2EALWAYS__EVENTUALLY,axiom,
! [A_27a: $tType,V0p: A_27a > $o,V1net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ! [V2x: A_27a] : ( V0p @ V2x )
=> ( c_2Ereal__topology_2Eeventually @ A_27a @ V0p @ V1net ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__AND,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1p: A_27a > $o,V2q: A_27a > $o] :
( ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V1p @ V3x ) @ ( V2q @ V3x ) )
@ V0net )
<=> ( ( c_2Ereal__topology_2Eeventually @ A_27a @ V1p @ V0net )
& ( c_2Ereal__topology_2Eeventually @ A_27a @ V2q @ V0net ) ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__MONO,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1p: A_27a > $o,V2q: A_27a > $o] :
( ( ! [V3x: A_27a] :
( ( V1p @ V3x )
=> ( V2q @ V3x ) )
& ( c_2Ereal__topology_2Eeventually @ A_27a @ V1p @ V0net ) )
=> ( c_2Ereal__topology_2Eeventually @ A_27a @ V2q @ V0net ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__MP,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1p: A_27a > $o,V2q: A_27a > $o] :
( ( ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V3x: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( V1p @ V3x ) @ ( V2q @ V3x ) )
@ V0net )
& ( c_2Ereal__topology_2Eeventually @ A_27a @ V1p @ V0net ) )
=> ( c_2Ereal__topology_2Eeventually @ A_27a @ V2q @ V0net ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__FALSE,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V1x: A_27a] : c_2Ebool_2EF
@ V0net )
= ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__TRUE,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V1x: A_27a] : c_2Ebool_2ET
@ V0net )
= c_2Ebool_2ET ) ).
thf(thm_2Ereal__topology_2ENOT__EVENTUALLY,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1p: A_27a > $o] :
( ( ! [V2x: A_27a] : ( (~) @ ( V1p @ V2x ) )
& ( (~) @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net ) ) )
=> ( (~) @ ( c_2Ereal__topology_2Eeventually @ A_27a @ V1p @ V0net ) ) ) ).
thf(thm_2Ereal__topology_2EEVENTUALLY__FORALL,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1p: A_27b > A_27a > $o,V2s: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V2s )
& ( (~)
@ ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) )
=> ( ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V3x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V4a: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27b @ V4a @ V2s ) @ ( V1p @ V4a @ V3x ) ) )
@ V0net )
<=> ! [V5a: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V5a @ V2s )
=> ( c_2Ereal__topology_2Eeventually @ A_27a @ ( V1p @ V5a ) @ V0net ) ) ) ) ).
thf(thm_2Ereal__topology_2EFORALL__EVENTUALLY,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1p: A_27b > A_27a > $o,V2s: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V2s )
& ( (~)
@ ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) )
=> ( ! [V3a: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3a @ V2s )
=> ( c_2Ereal__topology_2Eeventually @ A_27a @ ( V1p @ V3a ) @ V0net ) )
<=> ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V4x: A_27a] :
( c_2Ebool_2E_21 @ A_27b
@ ^ [V5a: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27b @ V5a @ V2s ) @ ( V1p @ V5a @ V4x ) ) )
@ V0net ) ) ) ).
thf(thm_2Ereal__topology_2ELIM,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1l: tyop_2Erealax_2Ereal,V2f: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2f @ V1l @ V0net )
<=> ( ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V0net )
| ! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4y: A_27a] :
( ? [V5x: A_27a] : ( c_2Ereal__topology_2Enetord @ A_27a @ V0net @ V5x @ V4y )
& ! [V6x: A_27a] :
( ( c_2Ereal__topology_2Enetord @ A_27a @ V0net @ V6x @ V4y )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V2f @ V6x ) @ V1l ) ) @ V3e ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__WITHIN__LE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2a ) @ V3s ) )
<=> ! [V4e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4e )
=> ? [V5d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5d )
& ! [V6x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V3s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6x @ V2a ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6x @ V2a ) ) @ V5d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V1l ) ) @ V4e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__WITHIN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2a ) @ V3s ) )
<=> ! [V4e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4e )
=> ? [V5d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5d )
& ! [V6x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V3s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6x @ V2a ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6x @ V2a ) ) @ V5d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V1l ) ) @ V4e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__AT__LE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Eat @ V2a ) )
<=> ! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4d )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x @ V2a ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x @ V2a ) ) @ V4d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V1l ) ) @ V3e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__AT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Eat @ V2a ) )
<=> ! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4d )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x @ V2a ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x @ V2a ) ) @ V4d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V1l ) ) @ V3e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__AT__INFINITY,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ c_2Ereal__topology_2Eat__infinity )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3b: tyop_2Erealax_2Ereal] :
! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__ge @ ( c_2Ereal_2Eabs @ V4x ) @ V3b )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V1l ) ) @ V2e ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__AT__INFINITY__POS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ c_2Ereal__topology_2Eat__infinity )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3b: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3b )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__ge @ ( c_2Ereal_2Eabs @ V4x ) @ V3b )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V1l ) ) @ V2e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__AT__POSINFINITY,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ c_2Ereal__topology_2Eat__posinfinity )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3b: tyop_2Erealax_2Ereal] :
! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__ge @ V4x @ V3b )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V1l ) ) @ V2e ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__AT__NEGINFINITY,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ c_2Ereal__topology_2Eat__neginfinity )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3b: tyop_2Erealax_2Ereal] :
! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V4x @ V3b )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V1l ) ) @ V2e ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__SEQUENTIALLY,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0s @ V1l @ c_2Ereal__topology_2Esequentially )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3N: tyop_2Enum_2Enum] :
! [V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V3N @ V4n )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0s @ V4n ) @ V1l ) ) @ V2e ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__EVENTUALLY,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V3x: A_27a] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V1f @ V3x ) @ V2l )
@ V0net )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V2l @ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__POSINFINITY__SEQUENTIALLY,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ c_2Ereal__topology_2Eat__posinfinity )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V2n: tyop_2Enum_2Enum] : ( V0f @ ( c_2Ereal_2Ereal__of__num @ V2n ) )
@ V1l
@ c_2Ereal__topology_2Esequentially ) ) ).
thf(thm_2Ereal__topology_2ELIM__INFINITY__POSINFINITY,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ c_2Ereal__topology_2Eat__infinity )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ c_2Ereal__topology_2Eat__posinfinity ) ) ).
thf(thm_2Ereal__topology_2ELIM__WITHIN__EMPTY,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2x ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__WITHIN__SUBSET,axiom,
! [V0t: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3a: tyop_2Erealax_2Ereal,V4s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V1f @ V2l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3a ) @ V4s ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0t @ V4s ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V1f @ V2l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3a ) @ V0t ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__UNION,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o,V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V2l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1x ) @ V3s ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V2l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1x ) @ V4t ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V2l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1x ) @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V3s @ V4t ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__UNION__UNIV,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o,V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V2l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1x ) @ V3s ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V2l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1x ) @ V4t ) )
& ( ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V3s @ V4t )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V2l @ ( c_2Ereal__topology_2Eat @ V1x ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__COMPOSE__WITHIN,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o,V4y: tyop_2Erealax_2Ereal,V5z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V4y @ V0net )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V6w: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1f @ V6w ) @ V3s ) @ ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V1f @ V6w ) @ V4y ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V2g @ V4y ) @ V5z ) ) )
@ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V2g @ V5z @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V4y ) @ V3s ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2g @ V1f ) @ V5z @ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__COMPOSE__AT,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal,V4z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V1f @ V3y @ V0net )
& ( c_2Ereal__topology_2Eeventually @ A_27a
@ ^ [V5w: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V1f @ V5w ) @ V3y ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V2g @ V3y ) @ V4z ) )
@ V0net )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V2g @ V4z @ ( c_2Ereal__topology_2Eat @ V3y ) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2g @ V1f ) @ V4z @ V0net ) ) ).
thf(thm_2Ereal__topology_2ELIM__AT__WITHIN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Eat @ V2a ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2a ) @ V3s ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__WITHIN__OPEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2a @ V3s )
& ( c_2Ereal__topology_2EOpen @ V3s ) )
=> ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2a ) @ V3s ) )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ V1l @ ( c_2Ereal__topology_2Eat @ V2a ) ) ) ) ).
thf(thm_2Ereal__topology_2EFROM__0,axiom,
( ( c_2Ereal__topology_2Efrom @ c_2Enum_2E0 )
= ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ).
thf(thm_2Ereal__topology_2EIN__FROM,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0m @ ( c_2Ereal__topology_2Efrom @ V1n ) )
= ( c_2Earithmetic_2E_3C_3D @ V1n @ V0m ) ) ).
thf(thm_2Ereal__topology_2EFROM__INTER__NUMSEG__GEN,axiom,
! [V0k: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ ( c_2Ereal__topology_2Efrom @ V0k ) @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Enum_2Enum > $o ) @ ( c_2Eprim__rec_2E_3C @ V1m @ V0k ) @ ( c_2Eiterate_2E_2E_2E @ V0k @ V2n ) @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) ) ) ).
thf(thm_2Ereal__topology_2EFROM__INTER__NUMSEG__MAX,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ ( c_2Ereal__topology_2Efrom @ V2p ) @ ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) )
= ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2EMAX @ V2p @ V0m ) @ V1n ) ) ).
thf(thm_2Ereal__topology_2EFROM__INTER__NUMSEG,axiom,
! [V0k: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ ( c_2Ereal__topology_2Efrom @ V0k ) @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V1n ) )
= ( c_2Eiterate_2E_2E_2E @ V0k @ V1n ) ) ).
thf(thm_2Ereal__topology_2EINFINITE__FROM,axiom,
! [V0n: tyop_2Enum_2Enum] : ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ ( c_2Ereal__topology_2Efrom @ V0n ) ) ) ).
thf(thm_2Ereal__topology_2EWLOG__LT,axiom,
! [V0P: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( ( ! [V1m: tyop_2Enum_2Enum] : ( V0P @ V1m @ V1m )
& ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( V0P @ V2m @ V3n )
= ( V0P @ V3n @ V2m ) )
& ! [V4m: tyop_2Enum_2Enum,V5n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V4m @ V5n )
=> ( V0P @ V4m @ V5n ) ) )
=> ! [V6m: tyop_2Enum_2Enum,V7y: tyop_2Enum_2Enum] : ( V0P @ V6m @ V7y ) ) ).
thf(thm_2Ereal__topology_2EWLOG__LE,axiom,
! [V0P: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( ( ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( V0P @ V1m @ V2n )
= ( V0P @ V2n @ V1m ) )
& ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V3m @ V4n )
=> ( V0P @ V3m @ V4n ) ) )
=> ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] : ( V0P @ V5m @ V6n ) ) ).
thf(thm_2Ereal__topology_2EREAL__LE__BETWEEN,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0a @ V1b )
<=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0a @ V2x )
& ( c_2Ereal_2Ereal__lte @ V2x @ V1b ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDS__LINEAR__0,axiom,
! [V0A: tyop_2Enum_2Enum,V1B: tyop_2Enum_2Enum] :
( ! [V2n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ V0A @ V2n ) @ V1B )
<=> ( V0A = c_2Enum_2E0 ) ) ).
thf(thm_2Ereal__topology_2EBOUNDS__LINEAR,axiom,
! [V0A: tyop_2Enum_2Enum,V1B: tyop_2Enum_2Enum,V2C: tyop_2Enum_2Enum] :
( ! [V3n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ V0A @ V3n ) @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V1B @ V3n ) @ V2C ) )
<=> ( c_2Earithmetic_2E_3C_3D @ V0A @ V1B ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__BIGINTER,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0f )
=> ( c_2Ereal__topology_2EClosed @ V1s ) )
=> ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t ) )
=> ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t ) )
=> ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__UNIV,axiom,
c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2ECLOSED__EMPTY,axiom,
c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2EOPEN__SUB__OPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ? [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V2t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V2t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2t @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
& ( c_2Ereal__topology_2EOpen @ V1t ) )
=> ( c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
= ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__REFL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V0s ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__REFL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V0s ) ).
thf(thm_2Ereal__topology_2ETOPSPACE__EUCLIDEAN__SUBTOPOLOGY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Etopspace @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) )
= V0s ) ).
thf(thm_2Ereal__topology_2ETOPSPACE__EUCLIDEAN,axiom,
( ( c_2Etopology_2Etopspace @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) ) ).
thf(thm_2Ereal__topology_2EOPEN__EXISTS,axiom,
! [A_27a: $tType,V0Q: A_27a > tyop_2Erealax_2Ereal > $o] :
( ! [V1a: A_27a] :
( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( V0Q @ V1a @ V2x ) ) ) )
=> ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V4a: A_27a] : ( V0Q @ V4a @ V3x ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__EXISTS__IN,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > tyop_2Erealax_2Ereal > $o] :
( ! [V2a: A_27a] :
( ( V0P @ V2a )
=> ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( V1Q @ V2a @ V3x ) ) ) ) )
=> ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V5a: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V0P @ V5a ) @ ( V1Q @ V5a @ V4x ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__BIGUNION,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0f )
=> ( c_2Ereal__topology_2EOpen @ V1s ) )
=> ( c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V0s )
& ( c_2Ereal__topology_2EOpen @ V1t ) )
=> ( c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__UNIV,axiom,
c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2EOPEN__EMPTY,axiom,
c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__UNION,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o,V2t: A_27a > $o,V3u: A_27a > $o] :
( ( ( c_2Etopology_2Eclosed__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V2t ) @ V1s )
& ( c_2Etopology_2Eclosed__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V3u ) @ V1s ) )
=> ( c_2Etopology_2Eclosed__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ ( c_2Epred__set_2EUNION @ A_27a @ V2t @ V3u ) ) @ V1s ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__UNION,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o,V2t: A_27a > $o,V3u: A_27a > $o] :
( ( ( c_2Etopology_2Eopen__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V2t ) @ V1s )
& ( c_2Etopology_2Eopen__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V3u ) @ V1s ) )
=> ( c_2Etopology_2Eopen__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ ( c_2Epred__set_2EUNION @ A_27a @ V2t @ V3u ) ) @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__IMP__SUBSET,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Etopology_2Eclosed__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V1s ) @ V2t )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V1s ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__IMP__SUBSET,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Etopology_2Eopen__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V1s ) @ V2t )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V1s ) ) ).
thf(thm_2Ereal__topology_2ESUBTOPOLOGY__UNIV,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V0top ) ).
thf(thm_2Ereal__topology_2ESUBTOPOLOGY__TOPSPACE,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) )
= V0top ) ).
thf(thm_2Ereal__topology_2ESUBTOPOLOGY__SUPERSET,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) @ V1s )
=> ( ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V1s )
= V0top ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__REFL,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1u: A_27a > $o] :
( ( c_2Etopology_2Eclosed__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V1u ) @ V1u )
= ( c_2Epred__set_2ESUBSET @ A_27a @ V1u @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__REFL,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1u: A_27a > $o] :
( ( c_2Etopology_2Eopen__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V1u ) @ V1u )
= ( c_2Epred__set_2ESUBSET @ A_27a @ V1u @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__EMPTY,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o] :
( ( c_2Etopology_2Eclosed__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1s )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__EMPTY,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o] :
( ( c_2Etopology_2Eopen__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1s )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1u: A_27a > $o,V2s: A_27a > $o] :
( ( c_2Etopology_2Eclosed__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V1u ) @ V2s )
<=> ? [V3t: A_27a > $o] :
( ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V3t )
& ( V2s
= ( c_2Epred__set_2EINTER @ A_27a @ V3t @ V1u ) ) ) ) ).
thf(thm_2Ereal__topology_2ETOPSPACE__SUBTOPOLOGY,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1u: A_27a > $o] :
( ( c_2Etopology_2Etopspace @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V1u ) )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) @ V1u ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1u: A_27a > $o,V2s: A_27a > $o] :
( ( c_2Etopology_2Eopen__in @ A_27a @ ( c_2Ereal__topology_2Esubtopology @ A_27a @ V0top @ V1u ) @ V2s )
<=> ? [V3t: A_27a > $o] :
( ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V3t )
& ( V2s
= ( c_2Epred__set_2EINTER @ A_27a @ V3t @ V1u ) ) ) ) ).
thf(thm_2Ereal__topology_2EISTOPLOGY__SUBTOPOLOGY,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1u: A_27a > $o] :
( c_2Etopology_2Eistopology @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V2s: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ ( c_2Epred__set_2EINTER @ A_27a @ V2s @ V1u ) @ ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V2s ) ) ) ) ).
thf(thm_2Ereal__topology_2EINTER__BIGUNION,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0s: ( A_27a > $o ) > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0s ) @ V1t )
= ( c_2Epred__set_2EBIGUNION @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V2x: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ ( c_2Epred__set_2EINTER @ A_27a @ V2x @ V1t ) @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2x @ V0s ) ) ) ) )
& ! [V3s: ( A_27b > $o ) > $o,V4t: A_27b > $o] :
( ( c_2Epred__set_2EINTER @ A_27b @ V4t @ ( c_2Epred__set_2EBIGUNION @ A_27b @ V3s ) )
= ( c_2Epred__set_2EBIGUNION @ A_27b
@ ( c_2Epred__set_2EGSPEC @ ( A_27b > $o ) @ ( A_27b > $o )
@ ^ [V5x: A_27b > $o] : ( c_2Epair_2E_2C @ ( A_27b > $o ) @ $o @ ( c_2Epred__set_2EINTER @ A_27b @ V4t @ V5x ) @ ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V5x @ V3s ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSET__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27b @ V1s @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2t ) )
<=> ? [V3u: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V3u @ V2t )
& ( V1s
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V3u ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIM__SUBSET__UNIV,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Ereal__topology_2Edim @ V0s ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Ereal__topology_2EDIM__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Ereal__topology_2Edim @ V0s ) @ ( c_2Ereal__topology_2Edim @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EDIM__UNIV,axiom,
( ( c_2Ereal__topology_2Edim @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Ereal__topology_2EHAS__SIZE__STDBASIS,axiom,
( c_2Ecardinal_2EHAS__SIZE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V0i: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V0i @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0i ) @ ( c_2Ereal_2Ereal__lte @ V0i @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
@ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__STDBASIS,axiom,
( ( c_2Ereal__topology_2Espan
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V0i: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V0i @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0i ) @ ( c_2Ereal_2Ereal__lte @ V0i @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
= ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2EDIM__LE__CARD,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Ereal__topology_2Edim @ V0s ) @ ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EDIM__UNIQUE,axiom,
! [V0n: tyop_2Enum_2Enum,V1v: tyop_2Erealax_2Ereal > $o,V2b: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2b @ V1v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1v @ ( c_2Ereal__topology_2Espan @ V2b ) )
& ( c_2Ereal__topology_2Eindependent @ V2b )
& ( c_2Ecardinal_2EHAS__SIZE @ tyop_2Erealax_2Ereal @ V2b @ V0n ) )
=> ( ( c_2Ereal__topology_2Edim @ V1v )
= V0n ) ) ).
thf(thm_2Ereal__topology_2EBASIS__HAS__SIZE__DIM,axiom,
! [V0v: tyop_2Erealax_2Ereal > $o,V1b: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Eindependent @ V1b )
& ( ( c_2Ereal__topology_2Espan @ V1b )
= V0v ) )
=> ( c_2Ecardinal_2EHAS__SIZE @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Ereal__topology_2Edim @ V0v ) ) ) ).
thf(thm_2Ereal__topology_2EBASIS__CARD__EQ__DIM,axiom,
! [V0v: tyop_2Erealax_2Ereal > $o,V1b: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1b @ V0v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0v @ ( c_2Ereal__topology_2Espan @ V1b ) )
& ( c_2Ereal__topology_2Eindependent @ V1b ) )
=> ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1b )
& ( ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V1b )
= ( c_2Ereal__topology_2Edim @ V0v ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__CARD__GE__DIM,axiom,
! [V0v: tyop_2Erealax_2Ereal > $o,V1b: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0v @ ( c_2Ereal__topology_2Espan @ V1b ) )
& ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1b ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Ereal__topology_2Edim @ V0v ) @ ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V1b ) ) ) ).
thf(thm_2Ereal__topology_2EINDEPENDENT__CARD__LE__DIM,axiom,
! [V0v: tyop_2Erealax_2Ereal > $o,V1b: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1b @ V0v )
& ( c_2Ereal__topology_2Eindependent @ V1b ) )
=> ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1b )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V1b ) @ ( c_2Ereal__topology_2Edim @ V0v ) ) ) ) ).
thf(thm_2Ereal__topology_2EBASIS__EXISTS,axiom,
! [V0v: tyop_2Erealax_2Ereal > $o] :
? [V1b: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1b @ V0v )
& ( c_2Ereal__topology_2Eindependent @ V1b )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0v @ ( c_2Ereal__topology_2Espan @ V1b ) )
& ( c_2Ecardinal_2EHAS__SIZE @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Ereal__topology_2Edim @ V0v ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__INJECTIVE__LEFT__INVERSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( V0f @ V1x )
= ( V0f @ V2y ) )
=> ( V1x = V2y ) ) )
=> ? [V3g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V3g )
& ( ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3g @ V0f )
= ( ^ [V4x: tyop_2Erealax_2Ereal] : V4x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__EQ__STDBASIS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Elinear @ V1g )
& ! [V2i: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2i )
& ( c_2Ereal_2Ereal__lte @ V2i @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( V0f @ V2i )
= ( V1g @ V2i ) ) ) )
=> ( V0f = V1g ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal > $o,V3s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Elinear @ V1g )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3s @ ( c_2Ereal__topology_2Espan @ V2b ) )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V2b )
=> ( ( V0f @ V4x )
= ( V1g @ V4x ) ) ) )
=> ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V3s )
=> ( ( V0f @ V5x )
= ( V1g @ V5x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__EQ__0,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal > $o,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2s @ ( c_2Ereal__topology_2Espan @ V1b ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1b )
=> ( ( V0f @ V3x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V2s )
=> ( ( V0f @ V4x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__EQ__0__SPAN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1b )
=> ( ( V0f @ V2x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Ereal__topology_2Espan @ V1b ) )
=> ( ( V0f @ V3x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__KERNEL,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( c_2Ereal__topology_2Esubspace
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V1x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__INDEPENDENT__EXTEND,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eindependent @ V1b )
=> ? [V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V2g )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1b )
=> ( ( V2g @ V3x )
= ( V0f @ V3x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__INDEPENDENT__EXTEND__LEMMA,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1b )
=> ( ( c_2Ereal__topology_2Eindependent @ V1b )
=> ? [V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Ereal__topology_2Espan @ V1b ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ ( c_2Ereal__topology_2Espan @ V1b ) ) )
=> ( ( V2g @ ( c_2Erealax_2Ereal__add @ V3x @ V4y ) )
= ( c_2Erealax_2Ereal__add @ ( V2g @ V3x ) @ ( V2g @ V4y ) ) ) )
& ! [V5x: tyop_2Erealax_2Ereal,V6c: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ ( c_2Ereal__topology_2Espan @ V1b ) )
=> ( ( V2g @ ( c_2Erealax_2Ereal__mul @ V6c @ V5x ) )
= ( c_2Erealax_2Ereal__mul @ V6c @ ( V2g @ V5x ) ) ) )
& ! [V7x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V1b )
=> ( ( V2g @ V7x )
= ( V0f @ V7x ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__BREAKDOWN__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1a @ V2s ) ) )
<=> ? [V3k: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ V0x @ ( c_2Erealax_2Ereal__mul @ V3k @ V1a ) ) @ ( c_2Ereal__topology_2Espan @ V2s ) ) ) ).
thf(thm_2Ereal__topology_2EMAXIMAL__INDEPENDENT__SUBSET,axiom,
! [V0v: tyop_2Erealax_2Ereal > $o] :
? [V1b: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1b @ V0v )
& ( c_2Ereal__topology_2Eindependent @ V1b )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0v @ ( c_2Ereal__topology_2Espan @ V1b ) ) ) ).
thf(thm_2Ereal__topology_2EMAXIMAL__INDEPENDENT__SUBSET__EXTEND,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1v: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1v )
& ( c_2Ereal__topology_2Eindependent @ V0s ) )
=> ? [V2b: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V2b )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2b @ V1v )
& ( c_2Ereal__topology_2Eindependent @ V2b )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1v @ ( c_2Ereal__topology_2Espan @ V2b ) ) ) ) ).
thf(thm_2Ereal__topology_2EINDEPENDENT__BOUND,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eindependent @ V0s )
=> ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V0s ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINDEPENDENT__SPAN__BOUND,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1t )
& ( c_2Ereal__topology_2Eindependent @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Espan @ V1t ) ) )
=> ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V0s ) @ ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2ECARD__STDBASIS,axiom,
( ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Ereal__topology_2EEXCHANGE__LEMMA,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1t )
& ( c_2Ereal__topology_2Eindependent @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Espan @ V1t ) ) )
=> ? [V2t_27: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ecardinal_2EHAS__SIZE @ tyop_2Erealax_2Ereal @ V2t_27 @ ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal @ V1t ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V2t_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2t_27 @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Espan @ V2t_27 ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__TRANS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Espan @ V2s ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ V2s ) ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal__topology_2Espan @ V2s ) ) ) ).
thf(thm_2Ereal__topology_2EIN__SPAN__DELETE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal__topology_2Espan @ V2s ) )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EDELETE @ tyop_2Erealax_2Ereal @ V2s @ V1b ) ) ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1b @ ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EDELETE @ tyop_2Erealax_2Ereal @ V2s @ V1b ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESPANNING__SUBSET__INDEPENDENT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s )
& ( c_2Ereal__topology_2Eindependent @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Espan @ V1t ) ) )
=> ( V0s = V1t ) ) ).
thf(thm_2Ereal__topology_2EINDEPENDENT__STDBASIS,axiom,
( c_2Ereal__topology_2Eindependent
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V0i: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V0i @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0i ) @ ( c_2Ereal_2Ereal__lte @ V0i @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EINDEPENDENT__SING,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eindependent @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
<=> ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal__topology_2EINDEPENDENT__EMPTY,axiom,
c_2Ereal__topology_2Eindependent @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2ESPAN__EMPTY,axiom,
( ( c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__INDUCT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1h: tyop_2Erealax_2Ereal > $o] :
( ( ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1h ) )
& ( c_2Ereal__topology_2Esubspace @ V1h ) )
=> ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Ereal__topology_2Espan @ V0s ) )
=> ( V1h @ V3x ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__CLAUSES,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ! [V1a: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V2s )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Ereal__topology_2Espan @ V2s ) ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2Espan @ V0s ) )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal,V5s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Ereal__topology_2Espan @ V5s ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ ( c_2Ereal__topology_2Espan @ V5s ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V3x @ V4y ) @ ( c_2Ereal__topology_2Espan @ V5s ) ) )
& ! [V6x: tyop_2Erealax_2Ereal,V7c: tyop_2Erealax_2Ereal,V8s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ ( c_2Ereal__topology_2Espan @ V8s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V7c @ V6x ) @ ( c_2Ereal__topology_2Espan @ V8s ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__SPAN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] : ( c_2Ereal__topology_2Esubspace @ ( c_2Ereal__topology_2Espan @ V0s ) ) ).
thf(thm_2Ereal__topology_2ESPAN__MONO,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Espan @ V0s ) @ ( c_2Ereal__topology_2Espan @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ESPAN__SPAN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Espan @ ( c_2Ereal__topology_2Espan @ V0s ) )
= ( c_2Ereal__topology_2Espan @ V0s ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__UNION__CHAIN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Esubspace @ V0s )
& ( c_2Ereal__topology_2Esubspace @ V1t )
& ( c_2Ereal__topology_2Esubspace @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) )
=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
| ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__INJECTIVE__0__SUBSPACE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Esubspace @ V1s ) )
=> ( ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1s )
& ( ( V0f @ V2x )
= ( V0f @ V3y ) ) )
=> ( V2x = V3y ) )
<=> ! [V4x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s )
& ( ( V0f @ V4x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( V4x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__BIGINTER,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0f )
=> ( c_2Ereal__topology_2Esubspace @ V1s ) )
=> ( c_2Ereal__topology_2Esubspace @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Esubspace @ V0s )
& ( c_2Ereal__topology_2Esubspace @ V1t ) )
=> ( c_2Ereal__topology_2Esubspace @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__TRIVIAL,axiom,
c_2Ereal__topology_2Esubspace @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__LINEAR__PREIMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Esubspace @ V1s ) )
=> ( c_2Ereal__topology_2Esubspace
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V2x ) @ V1s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Esubspace @ V1s ) )
=> ( c_2Ereal__topology_2Esubspace @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__SUM,axiom,
! [A_27a: $tType,V0s: tyop_2Erealax_2Ereal > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2t: A_27a > $o] :
( ( ( c_2Ereal__topology_2Esubspace @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V2t )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2t )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1f @ V3x ) @ V0s ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Eiterate_2ESum @ A_27a @ V2t @ V1f ) @ V0s ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Esubspace @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ V2s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) @ V2s ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Esubspace @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V1s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V0x ) @ V1s ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Esubspace @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V2s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V1c @ V0x ) @ V2s ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Esubspace @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1y @ V2s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ V2s ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__0,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esubspace @ V0s )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0s ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__IMP__NONEMPTY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Esubspace @ V0s )
=> ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSPACE__UNIV,axiom,
c_2Ereal__topology_2Esubspace @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2EBILINEAR__SUM__PARTIAL__PRE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Ebilinear @ V2h )
=> ( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V3m @ V4n )
@ ^ [V5k: tyop_2Enum_2Enum] : ( V2h @ ( V0f @ V5k ) @ ( c_2Ereal_2Ereal__sub @ ( V1g @ V5k ) @ ( V1g @ ( c_2Earithmetic_2E_2D @ V5k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2E_3C_3D @ V3m @ V4n )
@ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__sub @ ( V2h @ ( V0f @ ( c_2Earithmetic_2E_2B @ V4n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V1g @ V4n ) ) @ ( V2h @ ( V0f @ V3m ) @ ( V1g @ ( c_2Earithmetic_2E_2D @ V3m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V3m @ V4n )
@ ^ [V6k: tyop_2Enum_2Enum] : ( V2h @ ( c_2Ereal_2Ereal__sub @ ( V0f @ ( c_2Earithmetic_2E_2B @ V6k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V0f @ V6k ) ) @ ( V1g @ V6k ) ) ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__SUM__PARTIAL__SUC,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Ebilinear @ V2h )
=> ( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V3m @ V4n )
@ ^ [V5k: tyop_2Enum_2Enum] : ( V2h @ ( V0f @ V5k ) @ ( c_2Ereal_2Ereal__sub @ ( V1g @ ( c_2Earithmetic_2E_2B @ V5k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V1g @ V5k ) ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2E_3C_3D @ V3m @ V4n )
@ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__sub @ ( V2h @ ( V0f @ ( c_2Earithmetic_2E_2B @ V4n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V1g @ ( c_2Earithmetic_2E_2B @ V4n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( V2h @ ( V0f @ V3m ) @ ( V1g @ V3m ) ) )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V3m @ V4n )
@ ^ [V6k: tyop_2Enum_2Enum] : ( V2h @ ( c_2Ereal_2Ereal__sub @ ( V0f @ ( c_2Earithmetic_2E_2B @ V6k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V0f @ V6k ) ) @ ( V1g @ ( c_2Earithmetic_2E_2B @ V6k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__BOUNDED__POS,axiom,
! [V0h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0h )
=> ? [V1B: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1B )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0h @ V2x @ V3y ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__mul @ V1B @ ( c_2Ereal_2Eabs @ V2x ) ) @ ( c_2Ereal_2Eabs @ V3y ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__BOUNDED,axiom,
! [V0h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0h )
=> ? [V1B: tyop_2Erealax_2Ereal] :
! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0h @ V2x @ V3y ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__mul @ V1B @ ( c_2Ereal_2Eabs @ V2x ) ) @ ( c_2Ereal_2Eabs @ V3y ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__SUM,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o,V2g: A_27b > tyop_2Erealax_2Ereal,V3f: A_27a > tyop_2Erealax_2Ereal,V4h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Ebilinear @ V4h )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V0t ) )
=> ( ( V4h @ ( c_2Eiterate_2ESum @ A_27a @ V1s @ V3f ) @ ( c_2Eiterate_2ESum @ A_27b @ V0t @ V2g ) )
= ( c_2Eiterate_2ESum @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V1s @ V0t )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ tyop_2Erealax_2Ereal
@ ^ [V5i: A_27a,V6j: A_27b] : ( V4h @ ( V3f @ V5i ) @ ( V2g @ V6j ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__RSUB,axiom,
! [V0h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0h )
=> ( ( V0h @ V1x @ ( c_2Ereal_2Ereal__sub @ V2y @ V3z ) )
= ( c_2Ereal_2Ereal__sub @ ( V0h @ V1x @ V2y ) @ ( V0h @ V1x @ V3z ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__LSUB,axiom,
! [V0h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0h )
=> ( ( V0h @ ( c_2Ereal_2Ereal__sub @ V1x @ V2y ) @ V3z )
= ( c_2Ereal_2Ereal__sub @ ( V0h @ V1x @ V3z ) @ ( V0h @ V2y @ V3z ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__RZERO,axiom,
! [V0h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0h )
=> ( ( V0h @ V1x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__LZERO,axiom,
! [V0h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0h )
=> ( ( V0h @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__RNEG,axiom,
! [V0h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0h )
=> ( ( V0h @ V1x @ ( c_2Erealax_2Ereal__neg @ V2y ) )
= ( c_2Erealax_2Ereal__neg @ ( V0h @ V1x @ V2y ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__LNEG,axiom,
! [V0h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0h )
=> ( ( V0h @ ( c_2Erealax_2Ereal__neg @ V1x ) @ V2y )
= ( c_2Erealax_2Ereal__neg @ ( V0h @ V1x @ V2y ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__RMUL,axiom,
! [V0h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0h )
=> ( ( V0h @ V2x @ ( c_2Erealax_2Ereal__mul @ V1c @ V3y ) )
= ( c_2Erealax_2Ereal__mul @ V1c @ ( V0h @ V2x @ V3y ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__LMUL,axiom,
! [V0h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0h )
=> ( ( V0h @ ( c_2Erealax_2Ereal__mul @ V1c @ V2x ) @ V3y )
= ( c_2Erealax_2Ereal__mul @ V1c @ ( V0h @ V2x @ V3y ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__RADD,axiom,
! [V0h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0h )
=> ( ( V0h @ V1x @ ( c_2Erealax_2Ereal__add @ V2y @ V3z ) )
= ( c_2Erealax_2Ereal__add @ ( V0h @ V1x @ V2y ) @ ( V0h @ V1x @ V3z ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__LADD,axiom,
! [V0h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear @ V0h )
=> ( ( V0h @ ( c_2Erealax_2Ereal__add @ V1x @ V2y ) @ V3z )
= ( c_2Erealax_2Ereal__add @ ( V0h @ V1x @ V3z ) @ ( V0h @ V2y @ V3z ) ) ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__SWAP,axiom,
! [V0op: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebilinear
@ ^ [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] : ( V0op @ V2y @ V1x ) )
= ( c_2Ereal__topology_2Ebilinear @ V0op ) ) ).
thf(thm_2Ereal__topology_2ESYMMETRIC__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V2x ) @ V1s ) )
& ( c_2Ereal__topology_2Elinear @ V0f ) )
=> ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ V3x ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__BOUNDED__POS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ? [V1B: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1B )
& ! [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V2x ) ) @ ( c_2Erealax_2Ereal__mul @ V1B @ ( c_2Ereal_2Eabs @ V2x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__BOUNDED,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ? [V1B: tyop_2Erealax_2Ereal] :
! [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V2x ) ) @ ( c_2Erealax_2Ereal__mul @ V1B @ ( c_2Ereal_2Eabs @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__SUM__MUL,axiom,
! [A_27a: $tType,V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2c: A_27a > tyop_2Erealax_2Ereal,V3v: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) )
=> ( ( V0f
@ ( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V4i: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V2c @ V4i ) @ ( V3v @ V4i ) ) ) )
= ( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V5i: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V2c @ V5i ) @ ( V0f @ ( V3v @ V5i ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__SUM,axiom,
! [A_27a: $tType,V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Epred__set_2EFINITE @ A_27a @ V2s ) )
=> ( ( V0f @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1g ) )
= ( c_2Eiterate_2ESum @ A_27a @ V2s @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1g ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__SUB,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( ( V0f @ ( c_2Ereal_2Ereal__sub @ V1x @ V2y ) )
= ( c_2Ereal_2Ereal__sub @ ( V0f @ V1x ) @ ( V0f @ V2y ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__ADD,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( ( V0f @ ( c_2Erealax_2Ereal__add @ V1x @ V2y ) )
= ( c_2Erealax_2Ereal__add @ ( V0f @ V1x ) @ ( V0f @ V2y ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__NEG,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( ( V0f @ ( c_2Erealax_2Ereal__neg @ V1x ) )
= ( c_2Erealax_2Ereal__neg @ ( V0f @ V1x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__CMUL,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( ( V0f @ ( c_2Erealax_2Ereal__mul @ V1c @ V2x ) )
= ( c_2Erealax_2Ereal__mul @ V1c @ ( V0f @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__0,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( ( V0f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__MUL__COMPONENT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1v: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( c_2Ereal__topology_2Elinear
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ ( V0f @ V2x ) @ V1v ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__COMPOSE__SUM,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ! [V2a: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s )
=> ( c_2Ereal__topology_2Elinear @ ( V0f @ V2a ) ) ) )
=> ( c_2Ereal__topology_2Elinear
@ ^ [V3x: tyop_2Erealax_2Ereal] :
( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V4a: A_27a] : ( V0f @ V4a @ V3x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__NEGATION,axiom,
( c_2Ereal__topology_2Elinear
@ ^ [V0x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ V0x ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__ZERO,axiom,
( c_2Ereal__topology_2Elinear
@ ^ [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__ID,axiom,
( c_2Ereal__topology_2Elinear
@ ^ [V0x: tyop_2Erealax_2Ereal] : V0x ) ).
thf(thm_2Ereal__topology_2ELINEAR__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Elinear @ V1g ) )
=> ( c_2Ereal__topology_2Elinear @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__COMPOSE__SUB,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Elinear @ V1g ) )
=> ( c_2Ereal__topology_2Elinear
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub @ ( V0f @ V2x ) @ ( V1g @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__COMPOSE__ADD,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Elinear @ V0f )
& ( c_2Ereal__topology_2Elinear @ V1g ) )
=> ( c_2Ereal__topology_2Elinear
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ ( V0f @ V2x ) @ ( V1g @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__COMPOSE__NEG,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( c_2Ereal__topology_2Elinear
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ ( V0f @ V1x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__COMPOSE__CMUL,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( c_2Ereal__topology_2Elinear
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V1c @ ( V0f @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__SCALING,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2Elinear
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V0c @ V1x ) ) ).
thf(thm_2Ereal__topology_2EREAL__CHOOSE__DIST,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e )
=> ? [V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2y ) )
= V1e ) ) ).
thf(thm_2Ereal__topology_2EDIST__0,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
= ( c_2Ereal_2Eabs @ V0x ) )
& ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) )
= ( c_2Ereal_2Eabs @ V0x ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__EQ,axiom,
! [V0w: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0w @ V1x ) )
= ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2y @ V3z ) ) )
<=> ( ( c_2Ereal_2Epow @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0w @ V1x ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Ereal_2Epow @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2y @ V3z ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__POS__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__LE__0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ereal__topology_2EDIST__TRIANGLE__ADD__HALF,axiom,
! [V0e: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2x_27: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal,V4y_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2x_27 ) ) @ ( c_2Ereal_2E_2F @ V0e @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3y @ V4y_27 ) ) @ ( c_2Ereal_2E_2F @ V0e @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V1x @ V3y ) @ ( c_2Erealax_2Ereal__add @ V2x_27 @ V4y_27 ) ) ) @ V0e ) ) ).
thf(thm_2Ereal__topology_2EDIST__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V2c @ V0x ) @ ( c_2Erealax_2Ereal__mul @ V2c @ V1y ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V2c ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__TRIANGLE__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1x_27: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3y_27: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ V0x @ V2y ) @ ( c_2Erealax_2Ereal__add @ V1x_27 @ V3y_27 ) ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1x_27 ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2y @ V3y_27 ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__TRIANGLE__HALF__R,axiom,
! [V0e: tyop_2Erealax_2Ereal,V1x1: tyop_2Erealax_2Ereal,V2x2: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3y @ V1x1 ) ) @ ( c_2Ereal_2E_2F @ V0e @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3y @ V2x2 ) ) @ ( c_2Ereal_2E_2F @ V0e @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x1 @ V2x2 ) ) @ V0e ) ) ).
thf(thm_2Ereal__topology_2EDIST__TRIANGLE__HALF__L,axiom,
! [V0e: tyop_2Erealax_2Ereal,V1x1: tyop_2Erealax_2Ereal,V2x2: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x1 @ V3y ) ) @ ( c_2Ereal_2E_2F @ V0e @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x2 @ V3y ) ) @ ( c_2Ereal_2E_2F @ V0e @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x1 @ V2x2 ) ) @ V0e ) ) ).
thf(thm_2Ereal__topology_2EDIST__TRIANGLE__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal,V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2z ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1y @ V2z ) ) ) @ V3e )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) @ V3e ) ) ).
thf(thm_2Ereal__topology_2EDIST__TRIANGLE__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal,V3e: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2z ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1y @ V2z ) ) ) @ V3e )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) @ V3e ) ) ).
thf(thm_2Ereal__topology_2EDIST__NZ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V0x = V1y ) )
<=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__POS__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V0x = V1y ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__EQ__0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ereal__topology_2EDIST__TRIANGLE__ALT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1y @ V2z ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2z ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__TRIANGLE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V2z ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1y @ V2z ) ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__SYM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) )
= ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1y @ V0x ) ) ) ).
thf(thm_2Ereal__topology_2EDIST__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V0x ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal__topology_2EREAL__WLOG__LT,axiom,
! [V0P: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o] :
( ( ! [V1x: tyop_2Erealax_2Ereal] : ( V0P @ V1x @ V1x )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( V0P @ V2x @ V3y )
= ( V0P @ V3y @ V2x ) )
& ! [V4x: tyop_2Erealax_2Ereal,V5y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V4x @ V5y )
=> ( V0P @ V4x @ V5y ) ) )
=> ! [V6x: tyop_2Erealax_2Ereal,V7y: tyop_2Erealax_2Ereal] : ( V0P @ V6x @ V7y ) ) ).
thf(thm_2Ereal__topology_2EBIGUNION__MONO,axiom,
! [A_27a: $tType,V0t: ( A_27a > $o ) > $o,V1s: ( A_27a > $o ) > $o] :
( ! [V2x: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2x @ V1s )
=> ? [V3y: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3y @ V0t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2x @ V3y ) ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1s ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0t ) ) ) ).
thf(thm_2Ereal__topology_2EBIGUNION__MONO__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1g: A_27a > A_27b > $o,V2f: A_27a > A_27b > $o] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Epred__set_2ESUBSET @ A_27b @ ( V2f @ V3x ) @ ( V1g @ V3x ) ) )
=> ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Epred__set_2EBIGUNION @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( A_27b > $o ) @ V2f @ V0s ) ) @ ( c_2Epred__set_2EBIGUNION @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( A_27b > $o ) @ V1g @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2EDEPENDENT__CHOICE,axiom,
! [A_27a: $tType,V0P: tyop_2Enum_2Enum > A_27a > $o,V1R: tyop_2Enum_2Enum > A_27a > A_27a > $o] :
( ( ? [V2a: A_27a] : ( V0P @ c_2Enum_2E0 @ V2a )
& ! [V3n: tyop_2Enum_2Enum,V4x: A_27a] :
( ( V0P @ V3n @ V4x )
=> ? [V5y: A_27a] :
( ( V0P @ ( c_2Enum_2ESUC @ V3n ) @ V5y )
& ( V1R @ V3n @ V4x @ V5y ) ) ) )
=> ? [V6f: tyop_2Enum_2Enum > A_27a] :
( ! [V7n: tyop_2Enum_2Enum] : ( V0P @ V7n @ ( V6f @ V7n ) )
& ! [V8n: tyop_2Enum_2Enum] : ( V1R @ V8n @ ( V6f @ V8n ) @ ( V6f @ ( c_2Enum_2ESUC @ V8n ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EDEPENDENT__CHOICE__FIXED,axiom,
! [A_27a: $tType,V0P: tyop_2Enum_2Enum > A_27a > $o,V1R: tyop_2Enum_2Enum > A_27a > A_27a > $o,V2a: A_27a] :
( ( ( V0P @ c_2Enum_2E0 @ V2a )
& ! [V3n: tyop_2Enum_2Enum,V4x: A_27a] :
( ( V0P @ V3n @ V4x )
=> ? [V5y: A_27a] :
( ( V0P @ ( c_2Enum_2ESUC @ V3n ) @ V5y )
& ( V1R @ V3n @ V4x @ V5y ) ) ) )
=> ? [V6f: tyop_2Enum_2Enum > A_27a] :
( ( ( V6f @ c_2Enum_2E0 )
= V2a )
& ! [V7n: tyop_2Enum_2Enum] : ( V0P @ V7n @ ( V6f @ V7n ) )
& ! [V8n: tyop_2Enum_2Enum] : ( V1R @ V8n @ ( V6f @ V8n ) @ ( V6f @ ( c_2Enum_2ESUC @ V8n ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EEXISTS__IN__INSERT,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1a: A_27a,V2s: A_27a > $o] :
( ? [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Epred__set_2EINSERT @ A_27a @ V1a @ V2s ) )
& ( V0P @ V3x ) )
<=> ( ( V0P @ V1a )
| ? [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s )
& ( V0P @ V4x ) ) ) ) ).
thf(thm_2Ereal__topology_2EPERMUTES__INJECTIVE,axiom,
! [A_27a: $tType,V0p: A_27a > A_27a,V1s: A_27a > $o] :
( ( c_2Ereal__topology_2Epermutes @ A_27a @ V0p @ V1s )
=> ! [V2x: A_27a,V3y: A_27a] :
( ( ( V0p @ V2x )
= ( V0p @ V3y ) )
<=> ( V2x = V3y ) ) ) ).
thf(thm_2Ereal__topology_2EPERMUTES__IMAGE,axiom,
! [A_27a: $tType,V0p: A_27a > A_27a,V1s: A_27a > $o] :
( ( c_2Ereal__topology_2Epermutes @ A_27a @ V0p @ V1s )
=> ( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V0p @ V1s )
= V1s ) ) ).
thf(thm_2Ereal__topology_2EPAIRWISE__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27b > $o,V1r: A_27a > A_27a > $o,V2f: A_27b > A_27a] :
( ( c_2Ereal__topology_2Epairwise @ A_27a @ V1r @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ V2f @ V0s ) )
= ( c_2Ereal__topology_2Epairwise @ A_27b
@ ^ [V3x: A_27b,V4y: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ A_27a @ ( V2f @ V3x ) @ ( V2f @ V4y ) ) ) @ ( V1r @ ( V2f @ V3x ) @ ( V2f @ V4y ) ) )
@ V0s ) ) ).
thf(thm_2Ereal__topology_2EPAIRWISE__INSERT,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o,V1x: A_27a,V2s: A_27a > $o] :
( ( c_2Ereal__topology_2Epairwise @ A_27a @ V0r @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ V2s ) )
<=> ( ! [V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3y @ V2s )
& ( (~) @ ( V3y = V1x ) ) )
=> ( ( V0r @ V1x @ V3y )
& ( V0r @ V3y @ V1x ) ) )
& ( c_2Ereal__topology_2Epairwise @ A_27a @ V0r @ V2s ) ) ) ).
thf(thm_2Ereal__topology_2EPAIRWISE__MONO,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Ereal__topology_2Epairwise @ A_27a @ V0r @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V1s ) )
=> ( c_2Ereal__topology_2Epairwise @ A_27a @ V0r @ V2t ) ) ).
thf(thm_2Ereal__topology_2EPAIRWISE__SING,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o,V1x: A_27a] :
( ( c_2Ereal__topology_2Epairwise @ A_27a @ V0r @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= c_2Ebool_2ET ) ).
thf(thm_2Ereal__topology_2EPAIRWISE__EMPTY,axiom,
! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
( ( c_2Ereal__topology_2Epairwise @ A_27a @ V0r @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= c_2Ebool_2ET ) ).
thf(thm_2Ereal__topology_2EREAL__WLOG__LE,axiom,
! [V0P: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o] :
( ( ! [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( V0P @ V1x @ V2y )
= ( V0P @ V2y @ V1x ) )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V3x @ V4y )
=> ( V0P @ V3x @ V4y ) ) )
=> ! [V5x: tyop_2Erealax_2Ereal,V6y: tyop_2Erealax_2Ereal] : ( V0P @ V5x @ V6y ) ) ).
thf(thm_2Ereal__topology_2EIMAGE__SING,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1a: A_27a] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V1a @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( c_2Epred__set_2EINSERT @ A_27b @ ( V0f @ V1a ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ).
thf(thm_2Ereal__topology_2ESUM__ABS__TRIANGLE,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Ereal_2Ereal__lte
@ ( c_2Eiterate_2ESum @ A_27a @ V0s
@ ^ [V3a: A_27a] : ( c_2Ereal_2Eabs @ ( V1f @ V3a ) ) )
@ V2b ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Eiterate_2ESum @ A_27a @ V0s @ V1f ) ) @ V2b ) ) ).
thf(thm_2Ereal__topology_2ESUMS__SYM,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V1t ) ) ) ) )
= ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal )
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ $o )
@ ^ [V4y: tyop_2Erealax_2Ereal,V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ ( c_2Erealax_2Ereal__add @ V4y @ V5x ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V1t ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V0s ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUM__GP,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow @ V0x @ V3i ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Eprim__rec_2E_3C @ V2n @ V1m ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1m ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Epow @ V0x @ V1m ) @ ( c_2Ereal_2Epow @ V0x @ ( c_2Enum_2ESUC @ V2n ) ) ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUM__GP__MULTIPLIED,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
=> ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow @ V0x @ V3i ) ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Epow @ V0x @ V1m ) @ ( c_2Ereal_2Epow @ V0x @ ( c_2Enum_2ESUC @ V2n ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUM__GP__BASIC,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V1n )
@ ^ [V2i: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow @ V0x @ V2i ) ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ V0x @ ( c_2Enum_2ESUC @ V1n ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__INV__1__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Erealax_2Einv @ V0x ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
=> ( ! [V3k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3k )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3k ) ) )
<=> ! [V4u: tyop_2Erealax_2Ereal > $o,V5w: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V4u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5w @ V2t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5w ) ) ) )
@ V4u ) )
=> ? [V7v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V7v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5w @ V7v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V8x ) @ V7v ) ) ) )
@ V4u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3u: tyop_2Erealax_2Ereal > $o,V4w: tyop_2Erealax_2Ereal > $o] :
( ( ! [V5k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V5k )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5k ) ) )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4w @ V2t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V4w ) ) ) )
@ V3u ) )
=> ? [V7v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V7v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4w @ V7v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V8x ) @ V7v ) ) ) )
@ V3u ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__MAP__CLOSED__SUPERSET__PREIMAGE__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
=> ( ! [V3k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3k )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3k ) ) )
<=> ! [V4u: tyop_2Erealax_2Ereal > $o,V5w: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V4u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5w @ V2t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5w ) ) ) )
@ V4u ) )
=> ? [V7v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V7v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5w @ V7v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V8x ) @ V7v ) ) ) )
@ V4u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__MAP__CLOSED__SUPERSET__PREIMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3u: tyop_2Erealax_2Ereal > $o,V4w: tyop_2Erealax_2Ereal > $o] :
( ( ! [V5k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V5k )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5k ) ) )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4w @ V2t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V4w ) ) ) )
@ V3u ) )
=> ? [V7v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V7v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4w @ V7v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V8x ) @ V7v ) ) ) )
@ V3u ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__MAP__FROM__COMPOSITION__INJECTIVE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) @ V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3t ) @ V4u )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3t )
& ! [V5x: tyop_2Erealax_2Ereal,V6y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V3t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6y @ V3t )
& ( ( V1g @ V5x )
= ( V1g @ V6y ) ) )
=> ( V5x = V6y ) )
& ! [V7k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V7k )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V4u ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f ) @ V7k ) ) ) )
=> ! [V8k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V8k )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V8k ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__MAP__FROM__COMPOSITION__INJECTIVE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) @ V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3t ) @ V4u )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3t )
& ! [V5x: tyop_2Erealax_2Ereal,V6y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V3t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6y @ V3t )
& ( ( V1g @ V5x )
= ( V1g @ V6y ) ) )
=> ( V5x = V6y ) )
& ! [V7k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V7k )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V4u ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f ) @ V7k ) ) ) )
=> ! [V8k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V8k )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V8k ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__MAP__FROM__COMPOSITION__SURJECTIVE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s )
= V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3t ) @ V4u )
& ! [V5k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V5k )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V4u ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f ) @ V5k ) ) ) )
=> ! [V6k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ V6k )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V4u ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V6k ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__MAP__FROM__COMPOSITION__SURJECTIVE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s )
= V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3t ) @ V4u )
& ! [V5k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s ) @ V5k )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V4u ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f ) @ V5k ) ) ) )
=> ! [V6k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ V6k )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V4u ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V6k ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__MAP__IMP__CLOSED__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3u ) ) )
& ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V4u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V4u ) ) ) ) ) ) ) )
=> ! [V6u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V6u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V6u ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__MAP__IMP__OPEN__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3u ) ) )
& ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V4u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V4u ) ) ) ) ) ) ) )
=> ! [V6u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V6u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V6u ) ) ) ) ).
thf(thm_2Ereal__topology_2EQUOTIENT__MAP__CLOSED__MAP__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
& ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3u @ V2t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3u ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3u ) ) ) )
=> ( ! [V5k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V5k )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5k ) ) )
<=> ! [V6k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V6k )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V6k ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EQUOTIENT__MAP__OPEN__MAP__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
& ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3u @ V2t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3u ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3u ) ) ) )
=> ( ! [V5k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V5k )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5k ) ) )
<=> ! [V6k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V6k )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V6k ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__MAP__RESTRICT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3t_27: tyop_2Erealax_2Ereal > $o] :
( ( ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V4u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V4u ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3t_27 @ V2t ) )
=> ! [V5u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal
@ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V3t_27 ) ) ) ) )
@ V5u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t_27 ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5u ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__MAP__RESTRICT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3t_27: tyop_2Erealax_2Ereal > $o] :
( ( ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V4u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V4u ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3t_27 @ V2t ) )
=> ! [V5u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal
@ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V3t_27 ) ) ) ) )
@ V5u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t_27 ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5u ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__MAP__INTERIORS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V1s )
=> ( c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) )
<=> ! [V2s: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Einterior @ V2s ) ) @ ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__MAP__CLOSURES,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V1s )
=> ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) )
<=> ! [V2s: tyop_2Erealax_2Ereal > $o] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eclosure @ V2s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__MONOTONE__QUOTIENT__PREIMAGE__GEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3c: tyop_2Erealax_2Ereal > $o] :
( ( ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4u @ V2t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V4u ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V4u ) ) )
& ! [V6y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6y @ V2t )
=> ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ V6y ) ) ) ) ) )
& ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3c )
| ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3c ) )
& ( c_2Ereal__topology_2Econnected @ V3c ) )
=> ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V8x ) @ V3c ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__MONOTONE__QUOTIENT__PREIMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V2t )
& ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3u @ V2t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3u ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3u ) ) )
& ! [V5y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5y @ V2t )
=> ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5y ) ) ) ) ) )
& ( c_2Ereal__topology_2Econnected @ V2t ) )
=> ( c_2Ereal__topology_2Econnected @ V1s ) ) ).
thf(thm_2Ereal__topology_2EQUOTIENT__MAP__RESTRICT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3c: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
& ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4u @ V2t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V4u ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V4u ) ) )
& ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3c )
| ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3c ) ) )
=> ! [V6u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V6u @ V3c )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal
@ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ V3c ) ) ) ) )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x
@ ( c_2Ebool_2E_2F_5C
@ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V9x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V9x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V9x ) @ V3c ) ) ) ) )
@ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V8x ) @ V6u ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3c ) @ V6u ) ) ) ) ).
thf(thm_2Ereal__topology_2EQUOTIENT__MAP__FROM__SUBSET,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2t ) @ V3u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1s @ V2t )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s )
= V3u )
& ! [V4v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4v @ V3u )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V4v ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3u ) @ V4v ) ) ) )
=> ! [V6v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V6v @ V3u )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V2t ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ V6v ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3u ) @ V6v ) ) ) ) ).
thf(thm_2Ereal__topology_2EQUOTIENT__MAP__FROM__COMPOSITION,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) @ V3t )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3t ) @ V4u )
& ! [V5v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5v @ V4u )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f @ V6x ) @ V5v ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V4u ) @ V5v ) ) ) )
=> ! [V7v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V7v @ V4u )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V3t ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1g @ V8x ) @ V7v ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V4u ) @ V7v ) ) ) ) ).
thf(thm_2Ereal__topology_2EQUOTIENT__MAP__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) @ V3t )
& ! [V5v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5v @ V3t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5v ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ V5v ) ) )
& ! [V7v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V7v @ V4u )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V3t ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V1g @ V8x ) @ V7v ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V4u ) @ V7v ) ) ) )
=> ! [V9v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V9v @ V4u )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V10x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V10x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f @ V10x ) @ V9v ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V4u ) @ V9v ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIFT__TO__QUOTIENT__SPACE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s )
= V3t )
& ! [V5v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5v @ V3t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5v ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ V5v ) ) )
& ( c_2Ereal__topology_2Econtinuous__on @ V1h @ V2s )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1h @ V2s )
= V4u )
& ! [V7x: tyop_2Erealax_2Ereal,V8y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8y @ V2s )
& ( ( V0f @ V7x )
= ( V0f @ V8y ) ) )
=> ( ( V1h @ V7x )
= ( V1h @ V8y ) ) ) )
=> ? [V9g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous__on @ V9g @ V3t )
& ( ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V9g @ V3t )
= V4u )
& ! [V10x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V10x @ V2s )
=> ( ( V1h @ V10x )
= ( V9g @ ( V0f @ V10x ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EFUNCTION__FACTORS__LEFT__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27a > $o,V1f: A_27a > A_27b,V2g: A_27a > A_27c] :
( ! [V3x: A_27a,V4y: A_27a] :
( ( ( V0P @ V3x )
& ( V0P @ V4y )
& ( ( V2g @ V3x )
= ( V2g @ V4y ) ) )
=> ( ( V1f @ V3x )
= ( V1f @ V4y ) ) )
<=> ? [V5h: A_27c > A_27b] :
! [V6x: A_27a] :
( ( V0P @ V6x )
=> ( ( V1f @ V6x )
= ( V5h @ ( V2g @ V6x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__COMPOSE__QUOTIENT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) @ V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3t ) @ V4u )
& ! [V5v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5v @ V3t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5v ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ V5v ) ) )
& ( c_2Ereal__topology_2Econtinuous__on @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f ) @ V2s ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3t ) ) ).
thf(thm_2Ereal__topology_2EQUOTIENT__MAP__OPEN__CLOSED,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
=> ( ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3u @ V2t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3u ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3u ) ) )
<=> ! [V5u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5u @ V2t )
=> ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5u ) ) ) ) )
= ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V5u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__LEFT__INVERSE__IMP__QUOTIENT__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V2s )
=> ( ( V1g @ ( V0f @ V3x ) )
= V3x ) ) )
=> ! [V4u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4u @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V5x ) @ V4u ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) ) @ V4u ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__RIGHT__INVERSE__IMP__QUOTIENT__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) @ V3t )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3t ) @ V2s )
& ! [V4y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4y @ V3t )
=> ( ( V0f @ ( V1g @ V4y ) )
= V4y ) ) )
=> ! [V5u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5u @ V3t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V2s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5u ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ V5u ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__MAP__IMP__QUOTIENT__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V2t )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2t ) ) ) )
=> ! [V3t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3t @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3t ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) @ V3t ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__MAP__IMP__QUOTIENT__MAP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V2t )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2t ) ) ) )
=> ! [V3t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3t @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3t ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) @ V3t ) ) ) ) ).
thf(thm_2Ereal__topology_2EQUOTIENT__MAP__IMP__CONTINUOUS__CLOSED,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
& ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3u @ V2t )
=> ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3u ) ) ) ) )
= ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3u ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s ) ) ).
thf(thm_2Ereal__topology_2EQUOTIENT__MAP__IMP__CONTINUOUS__OPEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
& ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3u @ V2t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3u ) ) ) ) )
= ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3u ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s ) ) ).
thf(thm_2Ereal__topology_2EQUASICOMPACT__OPEN__CLOSED,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
=> ( ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3u @ V2t )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3u ) ) ) ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3u ) ) )
<=> ! [V5u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V5u @ V2t )
=> ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V6x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V6x ) @ V5u ) ) ) ) )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V5u ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econnected @ V1s )
& ( c_2Ereal__topology_2Elinear @ V0f ) )
=> ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__TRANSLATION__EQ,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V2x )
@ V1s ) )
= ( c_2Ereal__topology_2Econnected @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__TRANSLATION,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econnected @ V1s )
=> ( c_2Ereal__topology_2Econnected
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V2x )
@ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECONNECTED__CONTINUOUS__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2Econnected @ V1s ) )
=> ( c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__LINEAR__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V1s )
& ( c_2Ereal__topology_2Elinear @ V0f ) )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__TRANSLATION__EQ,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V2x )
@ V1s ) )
= ( c_2Ereal__topology_2Ecompact @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__TRANSLATION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V1a @ V2x )
@ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__CONTINUOUS__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2Ecompact @ V1s ) )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__DOT2,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V2s ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__DOT2,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V1f @ V0net )
& ( c_2Ereal__topology_2Econtinuous @ A_27a @ V2g @ V0net ) )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V1f @ V3x ) @ ( V2g @ V3x ) )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__DOT,axiom,
( c_2Ereal__topology_2Ebilinear
@ ^ [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__CONTINUOUS__ON__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V3s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V3s )
& ( c_2Ereal__topology_2Ebilinear @ V2h ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( V2h @ ( V0f @ V4x ) @ ( V1g @ V4x ) )
@ V3s ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__CONTINUOUS__COMPOSE,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: A_27a > tyop_2Erealax_2Ereal,V3h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V1f @ V0net )
& ( c_2Ereal__topology_2Econtinuous @ A_27a @ V2g @ V0net )
& ( c_2Ereal__topology_2Ebilinear @ V3h ) )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V4x: A_27a] : ( V3h @ ( V1f @ V4x ) @ ( V2g @ V4x ) )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENT__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( V0f @ V2x )
@ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__COMPONENT__COMPOSE,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2i: A_27b] :
( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V1f @ V0net )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V3x: A_27a] : ( V1f @ V3x )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__CONTINUOUS__ON__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Elinear @ V1g ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( V1g @ ( V0f @ V3x ) )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__CONTINUOUS__COMPOSE,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27a > tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V1f @ V0net )
& ( c_2Ereal__topology_2Elinear @ V2g ) )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V3x: A_27a] : ( V2g @ ( V1f @ V3x ) )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__CONTINUOUS__ON,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__CONTINUOUS__WITHIN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2x ) @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__CONTINUOUS__AT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V1a ) ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__LIM__0,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__topology_2Eat @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V2t )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V2t )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE__UNIV,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ! [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V2x ) )
& ( c_2Ereal__topology_2EClosed @ V1s ) )
=> ( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V1s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__OPEN__PREIMAGE__UNIV,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ! [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V2x ) )
& ( c_2Ereal__topology_2EOpen @ V1s ) )
=> ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V1s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2EClosed @ V1s )
& ( c_2Ereal__topology_2EClosed @ V2t ) )
=> ( c_2Ereal__topology_2EClosed
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2t ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__OPEN__PREIMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2EOpen @ V1s )
& ( c_2Ereal__topology_2EOpen @ V2t ) )
=> ( c_2Ereal__topology_2EOpen
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2t ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2EClosed @ V2t ) )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2t ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Ereal__topology_2EOpen @ V2t ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2t ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__IMP__CLOSED__IN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) @ V2t ) )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2t ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__GEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3u ) )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3u ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSED,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) @ V2t )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSED__GEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
=> ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3u )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3u ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__IMP__OPEN__IN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) @ V2t ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2t ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE__GEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3u ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3u ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) @ V2t )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2t ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN__GEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
=> ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V3u )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s )
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3u ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__OPEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V1x ) )
<=> ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V2t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V1x ) @ V2t ) )
=> ? [V3s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V3s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V3s )
& ! [V4x_27: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x_27 @ V3s )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V4x_27 ) @ V2t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__OPEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1x ) @ V2u ) )
<=> ! [V3t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EOpen @ V3t )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V1x ) @ V3t ) )
=> ? [V4s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V4s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V4s )
& ! [V5x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x_27 @ V4s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x_27 @ V2u ) )
=> ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V5x_27 ) @ V3t ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f ) @ V2s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V2x ) )
& ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V1g @ ( c_2Ereal__topology_2Eat @ ( V0f @ V2x ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f ) @ ( c_2Ereal__topology_2Eat @ V2x ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2x ) @ V3s ) )
& ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V1g @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ ( V0f @ V2x ) ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3s ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f ) @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2x ) @ V3s ) ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__ID,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( c_2Ereal__topology_2Euniformly__continuous__on
@ ^ [V1x: tyop_2Erealax_2Ereal] : V1x
@ V0s ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__ID,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( c_2Ereal__topology_2Econtinuous__on
@ ^ [V1x: tyop_2Erealax_2Ereal] : V1x
@ V0s ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__ID,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : V1x
@ ( c_2Ereal__topology_2Eat @ V0a ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__ID,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : V2x
@ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0a ) @ V1s ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUM,axiom,
! [A_27a: $tType,V0t: tyop_2Erealax_2Ereal > $o,V1f: A_27a > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3a: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3a @ V2s )
=> ( c_2Ereal__topology_2Euniformly__continuous__on @ ( V1f @ V3a ) @ V0t ) ) )
=> ( c_2Ereal__topology_2Euniformly__continuous__on
@ ^ [V4x: tyop_2Erealax_2Ereal] :
( c_2Eiterate_2ESum @ A_27a @ V2s
@ ^ [V5a: A_27a] : ( V1f @ V5a @ V4x ) )
@ V0t ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUB,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Euniformly__continuous__on @ V1g @ V2s ) )
=> ( c_2Ereal__topology_2Euniformly__continuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__ADD,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Euniformly__continuous__on @ V1g @ V2s ) )
=> ( c_2Ereal__topology_2Euniformly__continuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__NEG,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V1s )
=> ( c_2Ereal__topology_2Euniformly__continuous__on
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ ( V0f @ V2x ) )
@ V1s ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__VMUL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2v: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Euniformly__continuous__on @ V1c @ V0s )
=> ( c_2Ereal__topology_2Euniformly__continuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ ( V1c @ V3x ) @ V2v )
@ V0s ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CMUL,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V2s )
=> ( c_2Ereal__topology_2Euniformly__continuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V1c @ ( V0f @ V3x ) )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__MUL,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Euniformly__continuous__on @ V1g @ V2s )
& ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) )
& ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V2s ) ) )
=> ( c_2Ereal__topology_2Euniformly__continuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2EBILINEAR__UNIFORMLY__CONTINUOUS__ON__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V3s )
& ( c_2Ereal__topology_2Euniformly__continuous__on @ V1g @ V3s )
& ( c_2Ereal__topology_2Ebilinear @ V2h )
& ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3s ) )
& ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V3s ) ) )
=> ( c_2Ereal__topology_2Euniformly__continuous__on
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( V2h @ ( V0f @ V4x ) @ ( V1g @ V4x ) )
@ V3s ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__COMPOSE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Euniformly__continuous__on @ V1g @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2s ) ) )
=> ( c_2Ereal__topology_2Euniformly__continuous__on @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1g @ V0f ) @ V2s ) ) ).
thf(thm_2Ereal__topology_2ELINEAR__UNIFORMLY__CONTINUOUS__ON,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elinear @ V0f )
=> ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V1s ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CONST,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2Euniformly__continuous__on
@ ^ [V2x: tyop_2Erealax_2Ereal] : V1c
@ V0s ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__SUM,axiom,
! [A_27a: $tType,V0t: tyop_2Erealax_2Ereal > $o,V1f: A_27a > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3a: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3a @ V2s )
=> ( c_2Ereal__topology_2Econtinuous__on @ ( V1f @ V3a ) @ V0t ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V4x: tyop_2Erealax_2Ereal] :
( c_2Eiterate_2ESum @ A_27a @ V2s
@ ^ [V5a: A_27a] : ( V1f @ V5a @ V4x ) )
@ V0t ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__MIN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V2s ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emin @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__MAX,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V2s ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emax @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__ABS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs @ ( V0f @ V2x ) )
@ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__SUB,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V2s ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__ADD,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
& ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V2s ) )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__NEG,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ ( V0f @ V2x ) )
@ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CMUL,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s )
=> ( c_2Ereal__topology_2Econtinuous__on
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V1c @ ( V0f @ V3x ) )
@ V2s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__CONST,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1c: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2Econtinuous__on
@ ^ [V2x: tyop_2Erealax_2Ereal] : V1c
@ V0s ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__SUM,axiom,
! [A_27a: $tType,A_27b: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1f: A_27b > A_27a > tyop_2Erealax_2Ereal,V2s: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V2s )
& ! [V3a: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3a @ V2s )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a @ ( V1f @ V3a ) @ V0net ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V4x: A_27a] :
( c_2Eiterate_2ESum @ A_27b @ V2s
@ ^ [V5a: A_27b] : ( V1f @ V5a @ V4x ) )
@ V0net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__MIN,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V0f @ V2net )
& ( c_2Ereal__topology_2Econtinuous @ A_27a @ V1g @ V2net ) )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ereal_2Emin @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
@ V2net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__MAX,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V0f @ V2net )
& ( c_2Ereal__topology_2Econtinuous @ A_27a @ V1g @ V2net ) )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ereal_2Emax @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
@ V2net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ABS,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V0f @ V1net )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V2x: A_27a] : ( c_2Ereal_2Eabs @ ( V0f @ V2x ) )
@ V1net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__SUB,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V0f @ V2net )
& ( c_2Ereal__topology_2Econtinuous @ A_27a @ V1g @ V2net ) )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ereal_2Ereal__sub @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
@ V2net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ADD,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V0f @ V2net )
& ( c_2Ereal__topology_2Econtinuous @ A_27a @ V1g @ V2net ) )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__add @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
@ V2net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__NEG,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V0f @ V1net )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V2x: A_27a] : ( c_2Erealax_2Ereal__neg @ ( V0f @ V2x ) )
@ V1net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__CMUL,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( c_2Ereal__topology_2Econtinuous @ A_27a @ V0f @ V2net )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__mul @ V1c @ ( V0f @ V3x ) )
@ V2net ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__CONST,axiom,
! [A_27a: $tType,V0net: tyop_2Ereal__topology_2Enet @ A_27a,V1c: tyop_2Erealax_2Ereal] :
( c_2Ereal__topology_2Econtinuous @ A_27a
@ ^ [V2x: A_27a] : V1c
@ V0net ) ).
thf(thm_2Ereal__topology_2ELIM__CONTINUOUS__FUNCTION,axiom,
! [A_27a: $tType,V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1net: tyop_2Ereal__topology_2Enet @ A_27a,V2g: A_27a > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V3l ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a @ V2g @ V3l @ V1net ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a
@ ^ [V4x: A_27a] : ( V0f @ ( V2g @ V4x ) )
@ ( V0f @ V3l )
@ V1net ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SEQUENTIALLY,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V1s )
<=> ! [V2x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V4n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V2x @ V4n ) @ V1s )
& ! [V5n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V3y @ V5n ) @ V1s )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V6n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V2x @ V6n ) @ ( V3y @ V6n ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ c_2Ereal__topology_2Esequentially ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum
@ ^ [V7n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0f @ ( V2x @ V7n ) ) @ ( V0f @ ( V3y @ V7n ) ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__SEQUENTIALLY,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V2x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3a @ V1s )
& ! [V4n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V2x @ V4n ) @ V1s )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V2x @ V3a @ c_2Ereal__topology_2Esequentially ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2x ) @ ( V0f @ V3a ) @ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__SEQUENTIALLY,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V1a ) )
<=> ! [V2x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V2x @ V1a @ c_2Ereal__topology_2Esequentially )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V2x ) @ ( V0f @ V1a ) @ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__SEQUENTIALLY,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2a ) @ V1s ) )
<=> ! [V3x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V4n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V3x @ V4n ) @ V1s )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V3x @ V2a @ c_2Ereal__topology_2Esequentially ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V3x ) @ ( V0f @ V2a ) @ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2EFORALL__POS__MONO__1,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ! [V1d: tyop_2Erealax_2Ereal,V2e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V1d @ V2e )
& ( V0P @ V1d ) )
=> ( V0P @ V2e ) )
& ! [V3n: tyop_2Enum_2Enum] : ( V0P @ ( c_2Erealax_2Einv @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ V3n ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ! [V4e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4e )
=> ( V0P @ V4e ) ) ) ).
thf(thm_2Ereal__topology_2EISOMETRY__ON__IMP__CONTINUOUS__ON,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V0s ) )
=> ( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V1f @ V2x ) @ ( V1f @ V3y ) ) )
= ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V3y ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECONTRACTION__IMP__CONTINUOUS__ON,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V0s ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V1f @ V2x ) @ ( V1f @ V3y ) ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V3y ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V1f @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__FINITE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1s )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__NO__LIMPT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( (~)
@ ? [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V1s ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__EMPTY,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__SING,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econtinuous__on @ V0f @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V2s )
=> ( ( V0f @ V3x )
= ( V1g @ V3x ) ) )
& ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V2s ) )
=> ( c_2Ereal__topology_2Euniformly__continuous__on @ V1g @ V2s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V2s )
=> ( ( V0f @ V3x )
= ( V1g @ V3x ) ) )
& ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2s ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V1g @ V2s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__INTERIOR,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Ereal__topology_2Einterior @ V1s ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V2x ) ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUBSET,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V1s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2t @ V1s ) )
=> ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V2t ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__SUBSET,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2t @ V1s ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V2t ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__SUBSET,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3x ) @ V1s ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2t @ V1s ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3x ) @ V2t ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__EQ__CONTINUOUS__AT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V1s )
=> ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V2x ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ ( V0f @ V2x ) @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2x ) @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__ON__EQ__CONTINUOUS__WITHIN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
<=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2x ) @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__IMP__CONTINUOUS__ON,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V1s )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V2x ) ) )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__IMP__CONTINUOUS,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Euniformly__continuous__on @ V0f @ V1s )
=> ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__COMPARISON,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V1g @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3a ) @ V2s ) )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V2s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V3a ) @ ( V0f @ V4x ) ) ) @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V1g @ V3a ) @ ( V1g @ V4x ) ) ) ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3a ) @ V2s ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__BALL,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V1x ) )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3d )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V3d ) ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V1x ) @ V2e ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__BALL,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2x ) @ V1s ) )
<=> ! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4d )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2x @ V4d ) ) @ V1s ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0f @ V2x ) @ V3e ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Econtinuous__at,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V1f @ ( c_2Ereal__topology_2Eat @ V0x ) )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3d )
& ! [V4x_27: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4x_27 @ V0x ) ) @ V3d )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V1f @ V4x_27 ) @ ( V1f @ V0x ) ) ) @ V2e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2Econtinuous__within,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V2f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V0x ) @ V1s ) )
<=> ! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4d )
& ! [V5x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x_27 @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x_27 @ V0x ) ) @ V4d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V2f @ V5x_27 ) @ ( V2f @ V0x ) ) ) @ V3e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__SET__IMP,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Eeventually @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V3t ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V2s ) )
@ ( c_2Ereal__topology_2Eat @ V1a ) )
& ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1a ) @ V2s ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1a ) @ V3t ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__OPEN__IN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o,V4a: tyop_2Erealax_2Ereal] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V3t ) @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4a @ V2s )
& ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V2s )
=> ( ( V0f @ V5x )
= ( V1g @ V5x ) ) )
& ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V4a ) @ V3t ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V1g @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V4a ) @ V3t ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__OPEN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3a: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EOpen @ V2s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3a @ V2s )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V2s )
=> ( ( V0f @ V4x )
= ( V1g @ V4x ) ) )
& ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V3a ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V1g @ ( c_2Ereal__topology_2Eat @ V3a ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__AT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3d )
& ! [V4x_27: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4x_27 @ V2x ) ) @ V3d )
=> ( ( V0f @ V4x_27 )
= ( V1g @ V4x_27 ) ) )
& ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V2x ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V1g @ ( c_2Ereal__topology_2Eat @ V2x ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o,V3x: tyop_2Erealax_2Ereal,V4d: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4d )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V2s )
& ! [V5x_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x_27 @ V2s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5x_27 @ V3x ) ) @ V4d ) )
=> ( ( V0f @ V5x_27 )
= ( V1g @ V5x_27 ) ) )
& ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3x ) @ V2s ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V1g @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V3x ) @ V2s ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__CLOSED__NONTRIVIAL,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V2s )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1a @ V2s ) ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1a ) @ V2s ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT__WITHIN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V1x ) )
=> ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V1x ) @ V2s ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__AT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Ereal__topology_2Eat @ V1x ) )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V0f @ ( V0f @ V1x ) @ ( c_2Ereal__topology_2Eat @ V1x ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__WITHIN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econtinuous @ tyop_2Erealax_2Ereal @ V1f @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2x ) @ V0s ) )
= ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Erealax_2Ereal @ V1f @ ( V1f @ V2x ) @ ( c_2Ereal__topology_2Ewithin @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eat @ V2x ) @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECONTINUOUS__TRIVIAL__LIMIT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1net: tyop_2Ereal__topology_2Enet @ A_27a] :
( ( c_2Ereal__topology_2Etrivial__limit @ A_27a @ V1net )
=> ( c_2Ereal__topology_2Econtinuous @ A_27a @ V0f @ V1net ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CAUCHY__IMP__UNIFORMLY__CONVERGENT,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: tyop_2Enum_2Enum > A_27a > tyop_2Erealax_2Ereal,V2l: A_27a > tyop_2Erealax_2Ereal] :
( ( ! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4N: tyop_2Enum_2Enum] :
! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum,V7x: A_27a] :
( ( ( c_2Earithmetic_2E_3C_3D @ V4N @ V5m )
& ( c_2Earithmetic_2E_3C_3D @ V4N @ V6n )
& ( V0P @ V7x ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V1s @ V5m @ V7x ) @ ( V1s @ V6n @ V7x ) ) ) @ V3e ) ) )
& ! [V8x: A_27a] :
( ( V0P @ V8x )
=> ! [V9e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V9e )
=> ? [V10N: tyop_2Enum_2Enum] :
! [V11n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V10N @ V11n )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V1s @ V11n @ V8x ) @ ( V2l @ V8x ) ) ) @ V9e ) ) ) ) )
=> ! [V12e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V12e )
=> ? [V13N: tyop_2Enum_2Enum] :
! [V14n: tyop_2Enum_2Enum,V15x: A_27a] :
( ( ( c_2Earithmetic_2E_3C_3D @ V13N @ V14n )
& ( V0P @ V15x ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V1s @ V14n @ V15x ) @ ( V2l @ V15x ) ) ) @ V12e ) ) ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONVERGENT__EQ__CAUCHY__ALT,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: tyop_2Enum_2Enum > A_27a > tyop_2Erealax_2Ereal] :
( ? [V2l: A_27a > tyop_2Erealax_2Ereal] :
! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4N: tyop_2Enum_2Enum] :
! [V5n: tyop_2Enum_2Enum,V6x: A_27a] :
( ( ( c_2Earithmetic_2E_3C_3D @ V4N @ V5n )
& ( V0P @ V6x ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V1s @ V5n @ V6x ) @ ( V2l @ V6x ) ) ) @ V3e ) ) )
<=> ! [V7e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V7e )
=> ? [V8N: tyop_2Enum_2Enum] :
! [V9m: tyop_2Enum_2Enum,V10n: tyop_2Enum_2Enum,V11x: A_27a] :
( ( ( c_2Earithmetic_2E_3C_3D @ V8N @ V9m )
& ( c_2Earithmetic_2E_3C_3D @ V8N @ V10n )
& ( c_2Eprim__rec_2E_3C @ V9m @ V10n )
& ( V0P @ V11x ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V1s @ V9m @ V11x ) @ ( V1s @ V10n @ V11x ) ) ) @ V7e ) ) ) ) ).
thf(thm_2Ereal__topology_2EUNIFORMLY__CONVERGENT__EQ__CAUCHY,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: tyop_2Enum_2Enum > A_27a > tyop_2Erealax_2Ereal] :
( ? [V2l: A_27a > tyop_2Erealax_2Ereal] :
! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4N: tyop_2Enum_2Enum] :
! [V5n: tyop_2Enum_2Enum,V6x: A_27a] :
( ( ( c_2Earithmetic_2E_3C_3D @ V4N @ V5n )
& ( V0P @ V6x ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V1s @ V5n @ V6x ) @ ( V2l @ V6x ) ) ) @ V3e ) ) )
<=> ! [V7e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V7e )
=> ? [V8N: tyop_2Enum_2Enum] :
! [V9m: tyop_2Enum_2Enum,V10n: tyop_2Enum_2Enum,V11x: A_27a] :
( ( ( c_2Earithmetic_2E_3C_3D @ V8N @ V9m )
& ( c_2Earithmetic_2E_3C_3D @ V8N @ V10n )
& ( V0P @ V11x ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V1s @ V9m @ V11x ) @ ( V1s @ V10n @ V11x ) ) ) @ V7e ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__NEST,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > $o] :
( ( ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2Ecompact @ ( V0s @ V1n ) )
& ( (~)
@ ( ( V0s @ V1n )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
& ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V2m @ V3n )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0s @ V3n ) @ ( V0s @ V2m ) ) ) )
=> ( (~)
@ ( ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ tyop_2Enum_2Enum
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( V0s @ V4n ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__CHAIN,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0f )
=> ( ( c_2Ereal__topology_2Ecompact @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V2s: tyop_2Erealax_2Ereal > $o,V3t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V0f )
& ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V3t @ V0f ) )
=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2s @ V3t )
| ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3t @ V2s ) ) ) )
=> ( (~)
@ ( ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0f )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__CLOSED__CHAIN,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o,V1b: tyop_2Erealax_2Ereal > $o] :
( ( ! [V2s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2s @ V0f )
=> ( ( c_2Ereal__topology_2EClosed @ V2s )
& ( (~)
@ ( V2s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
& ! [V3s: tyop_2Erealax_2Ereal > $o,V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V3s @ V0f )
& ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V4t @ V0f ) )
=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3s @ V4t )
| ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4t @ V3s ) ) )
& ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1b @ V0f )
& ( c_2Ereal__topology_2Ebounded__def @ V1b ) )
=> ( (~)
@ ( ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0f )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EDECREASING__CLOSED__NEST__SING,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > $o] :
( ( ! [V1n: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2EClosed @ ( V0s @ V1n ) )
& ! [V2n: tyop_2Enum_2Enum] :
( (~)
@ ( ( V0s @ V2n )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V3m @ V4n )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0s @ V4n ) @ ( V0s @ V3m ) ) )
& ! [V5e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5e )
=> ? [V6n: tyop_2Enum_2Enum] :
! [V7x: tyop_2Erealax_2Ereal,V8y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ ( V0s @ V6n ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8y @ ( V0s @ V6n ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V7x @ V8y ) ) @ V5e ) ) ) )
=> ? [V9a: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o )
@ ^ [V10t: tyop_2Erealax_2Ereal > $o] :
( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ V10t
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V11n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ V10t @ ( V0s @ V11n ) ) ) ) ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V9a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EDECREASING__CLOSED__NEST,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > $o] :
( ( ! [V1n: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2EClosed @ ( V0s @ V1n ) )
& ! [V2n: tyop_2Enum_2Enum] :
( (~)
@ ( ( V0s @ V2n )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V3m @ V4n )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0s @ V4n ) @ ( V0s @ V3m ) ) )
& ! [V5e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5e )
=> ? [V6n: tyop_2Enum_2Enum] :
! [V7x: tyop_2Erealax_2Ereal,V8y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ ( V0s @ V6n ) )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8y @ ( V0s @ V6n ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V7x @ V8y ) ) @ V5e ) ) ) )
=> ? [V9a: tyop_2Erealax_2Ereal] :
! [V10n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9a @ ( V0s @ V10n ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__CLOSED__NEST,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > $o] :
( ( ! [V1n: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2EClosed @ ( V0s @ V1n ) )
& ! [V2n: tyop_2Enum_2Enum] :
( (~)
@ ( ( V0s @ V2n )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V3m @ V4n )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V0s @ V4n ) @ ( V0s @ V3m ) ) )
& ( c_2Ereal__topology_2Ebounded__def @ ( V0s @ c_2Enum_2E0 ) ) )
=> ? [V5a: tyop_2Erealax_2Ereal] :
! [V6n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5a @ ( V0s @ V6n ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__FIP,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V0f )
=> ( c_2Ereal__topology_2Ecompact @ V1t ) )
& ! [V2f_27: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V2f_27 )
& ( c_2Epred__set_2ESUBSET @ ( tyop_2Erealax_2Ereal > $o ) @ V2f_27 @ V0f ) )
=> ( (~)
@ ( ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V2f_27 )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ( (~)
@ ( ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0f )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__FIP,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V0f )
=> ( c_2Ereal__topology_2EClosed @ V1t ) )
& ? [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2t @ V0f )
& ( c_2Ereal__topology_2Ebounded__def @ V2t ) )
& ! [V3f_27: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 )
& ( c_2Epred__set_2ESUBSET @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 @ V0f ) )
=> ( (~)
@ ( ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V3f_27 )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ( (~)
@ ( ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0f )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IMP__FIP__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2t @ V1f )
=> ( c_2Ereal__topology_2Ecompact @ V2t ) )
& ! [V3f_27: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 )
& ( c_2Epred__set_2ESUBSET @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 @ V1f ) )
=> ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V3f_27 ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V1f ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IMP__FIP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2t @ V1f )
=> ( c_2Ereal__topology_2EClosed @ V2t ) )
& ? [V3t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V3t @ V1f )
& ( c_2Ereal__topology_2Ebounded__def @ V3t ) )
& ! [V4f_27: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V4f_27 )
& ( c_2Epred__set_2ESUBSET @ ( tyop_2Erealax_2Ereal > $o ) @ V4f_27 @ V1f ) )
=> ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V4f_27 ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V1f ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__IMP__FIP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2t @ V1f )
=> ( c_2Ereal__topology_2EClosed @ V2t ) )
& ! [V3f_27: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 )
& ( c_2Epred__set_2ESUBSET @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 @ V1f ) )
=> ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V3f_27 ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ( (~)
@ ( ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V1f ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Efrontier @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__SPHERE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EClosed @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SPHERE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ebounded__def @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__SPHERE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecompact @ ( c_2Ereal__topology_2Esphere @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1r ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__DIFF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2EOpen @ V1t ) )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EDIFF @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__BIGUNION,axiom,
! [V0s: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V0s )
& ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1t @ V0s )
=> ( c_2Ereal__topology_2Ecompact @ V1t ) ) )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__BIGINTER__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
<=> ! [V1e: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e ) ) @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__DELETE,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal > $o,V2a: tyop_2Erealax_2Ereal] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ V1s )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ ( c_2Epred__set_2EDELETE @ tyop_2Erealax_2Ereal @ V1s @ V2a ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__DELETE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EDELETE @ tyop_2Erealax_2Ereal @ V0s @ V1x ) ) ) ).
thf(thm_2Ereal__topology_2EFRONTIER__SUBSET__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Efrontier @ V0s ) @ V0s ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__FRONTIER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ( c_2Ereal__topology_2Ebounded__def @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__FRONTIER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__FRONTIER__BOUNDED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Ereal__topology_2Efrontier @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__CBALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecompact @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__INSERT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V1s )
=> ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSURE__SING,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__SING,axiom,
! [V0u: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0u ) @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1x @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
= ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0u ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__INSERT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V1s )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2EFINITE__IMP__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
=> ( c_2Ereal__topology_2Ecompact @ V0s ) ) ).
thf(thm_2Ereal__topology_2EFINITE__IMP__CLOSED__IN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
=> ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s ) ) ).
thf(thm_2Ereal__topology_2EFINITE__IMP__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
=> ( c_2Ereal__topology_2EClosed @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__BIGINTER,axiom,
! [V0f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0f )
=> ( c_2Ereal__topology_2Ecompact @ V1s ) )
& ( (~)
@ ( V0f
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Erealax_2Ereal > $o ) ) ) ) )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0f ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__INTER__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t ) )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__INTER__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2EClosed @ V1t ) )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t ) )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t ) )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__EMPTY,axiom,
c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2EBOUNDED__EQ__BOLZANO__WEIERSTRASS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
<=> ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s )
& ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1t ) ) )
=> ? [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s ) ) )
=> ? [V1x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Elimit__point__of @ V1x @ V0s ) ) ).
thf(thm_2Ereal__topology_2EDISCRETE__BOUNDED__IMP__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V0s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V3y @ V2x ) ) @ V1e ) )
=> ( V3y = V2x ) )
& ( c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s ) ) ).
thf(thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__CONTRAPOS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( (~) @ ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V1t ) ) ) )
=> ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1t ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ ( c_2Ereal__topology_2Eclosure @ V0s ) )
= ( c_2Ereal__topology_2Ebounded__def @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__EQ__HEINE__BOREL__SUBTOPOLOGY,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
<=> ! [V1f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2t @ V1f )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V2t ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V1f ) ) )
=> ? [V3f_27: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( c_2Epred__set_2ESUBSET @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 @ V1f )
& ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V3f_27 ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__COMPACT__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t )
<=> ( ( c_2Ereal__topology_2Ecompact @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IN__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t ) )
=> ( c_2Ereal__topology_2Ecompact @ V1t ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__SEQUENCE__WITH__LIMIT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0f @ V1l @ c_2Ereal__topology_2Esequentially )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V1l @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__IMP__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ( c_2Ereal__topology_2EClosed @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__IMP__BOUNDED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ( c_2Ereal__topology_2Ebounded__def @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__EQ__BOUNDED__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
<=> ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2EClosed @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__EQ__BOLZANO__WEIERSTRASS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
<=> ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1t ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s ) )
=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V1t ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__EQ__HEINE__BOREL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
<=> ! [V1f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2t @ V1f )
=> ( c_2Ereal__topology_2EOpen @ V2t ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V1f ) ) )
=> ? [V3f_27: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( c_2Epred__set_2ESUBSET @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 @ V1f )
& ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V3f_27 ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__IMP__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1t ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s ) )
=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V1t ) ) )
=> ( c_2Ereal__topology_2EClosed @ V0s ) ) ).
thf(thm_2Ereal__topology_2ESEQUENCE__UNIQUE__LIMPT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2l_27: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0f @ V1l @ c_2Ereal__topology_2Esequentially )
& ( c_2Ereal__topology_2Elimit__point__of @ V2l_27
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3y: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3y
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V3y @ ( V0f @ V4n ) ) ) ) ) ) )
=> ( V2l_27 = V1l ) ) ).
thf(thm_2Ereal__topology_2ELIMPT__OF__SEQUENCE__SUBSEQUENCE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elimit__point__of @ V1l @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
=> ? [V2r: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3m @ V4n )
=> ( c_2Eprim__rec_2E_3C @ ( V2r @ V3m ) @ ( V2r @ V4n ) ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ V0f @ V2r ) @ V1l @ c_2Ereal__topology_2Esequentially ) ) ) ).
thf(thm_2Ereal__topology_2ELE__1,axiom,
( ! [V0n: tyop_2Enum_2Enum] :
( ( (~) @ ( V0n = c_2Enum_2E0 ) )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n ) )
& ! [V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( V1n = c_2Enum_2E0 ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1n ) )
& ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
=> ( (~) @ ( V2n = c_2Enum_2E0 ) ) )
& ! [V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V3n )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V3n ) )
& ! [V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V4n )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V4n ) )
& ! [V5n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V5n )
=> ( (~) @ ( V5n = c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal__topology_2ESEQUENCE__INFINITE__LEMMA,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( ! [V2n: tyop_2Enum_2Enum] :
( (~)
@ ( ( V0f @ V2n )
= V1l ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0f @ V1l @ c_2Ereal__topology_2Esequentially ) )
=> ( (~)
@ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3y: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3y
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V3y @ ( V0f @ V4n ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__INF__LT__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Eiterate_2Einf @ V0s ) @ V1a )
<=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Erealax_2Ereal__lt @ V2x @ V1a ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LT__INF__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Erealax_2Ereal__lt @ V1a @ ( c_2Eiterate_2Einf @ V0s ) )
<=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Erealax_2Ereal__lt @ V1a @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__INF__LE__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2Einf @ V0s ) @ V1a )
<=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ereal_2Ereal__lte @ V2x @ V1a ) ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__LE__INF__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ V1a @ ( c_2Eiterate_2Einf @ V0s ) )
<=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V1a @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2EINF__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Eiterate_2Einf @ V0s ) @ V0s )
& ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2Einf @ V0s ) @ V1x ) ) ) ) ).
thf(thm_2Ereal__topology_2EINF__FINITE__LEMMA,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1b @ V0s )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V1b @ V2x ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__IMP__BOUNDED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ! [V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1t ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s ) )
=> ? [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Elimit__point__of @ V2x @ V1t ) )
=> ( c_2Ereal__topology_2Ebounded__def @ V0s ) ) ).
thf(thm_2Ereal__topology_2EHEINE__BOREL__IMP__BOLZANO__WEIERSTRASS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ! [V1f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2t @ V1f )
=> ( c_2Ereal__topology_2EOpen @ V2t ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V1f ) ) )
=> ? [V3f_27: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( c_2Epred__set_2ESUBSET @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 @ V1f )
& ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V3f_27 ) ) ) )
=> ! [V4t: tyop_2Erealax_2Ereal > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V4t ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4t @ V0s ) )
=> ? [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V0s )
& ( c_2Ereal__topology_2Elimit__point__of @ V5x @ V4t ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__IMP__HEINE__BOREL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ! [V1f: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2t @ V1f )
=> ( c_2Ereal__topology_2EOpen @ V2t ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V1f ) ) )
=> ? [V3f_27: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( c_2Epred__set_2ESUBSET @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 @ V1f )
& ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o ) @ V3f_27 )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V3f_27 ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EHEINE__BOREL__LEMMA,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ! [V1t: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V1t ) )
& ! [V2b: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2b @ V1t )
=> ( c_2Ereal__topology_2EOpen @ V2b ) ) )
=> ? [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
=> ? [V5b: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V5b @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4x @ V3e ) ) @ V5b ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__IMP__TOTALLY__BOUNDED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ! [V1e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e )
=> ? [V2k: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V2k )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2k @ V0s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s
@ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ ( tyop_2Erealax_2Ereal > $o )
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V3x @ V1e ) )
@ V2k ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONVERGENT__IMP__BOUNDED,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0s @ V1l @ c_2Ereal__topology_2Esequentially )
=> ( c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONVERGENT__EQ__CAUCHY,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ? [V1l: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0s @ V1l @ c_2Ereal__topology_2Esequentially )
<=> ( c_2Ereal__topology_2Ecauchy @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECOMPLETE__EQ__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecomplete @ V0s )
= ( c_2Ereal__topology_2EClosed @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECOMPLETE__UNIV,axiom,
c_2Ereal__topology_2Ecomplete @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2ECOMPACT__IMP__COMPLETE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ( c_2Ereal__topology_2Ecomplete @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECAUCHY__IMP__BOUNDED,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecauchy @ V0s )
=> ( c_2Ereal__topology_2Ebounded__def
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1y: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1y
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V1y @ ( V0s @ V2n ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2EUPPER__BOUND__FINITE__SET__REAL,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ? [V2a: tyop_2Erealax_2Ereal] :
! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V3x ) @ V2a ) ) ) ).
thf(thm_2Ereal__topology_2EGREATER__EQ__REFL,axiom,
! [V0m: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_3D @ V0m @ V0m ) ).
thf(thm_2Ereal__topology_2ECONVERGENT__IMP__CAUCHY,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0s @ V1l @ c_2Ereal__topology_2Esequentially )
=> ( c_2Ereal__topology_2Ecauchy @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECAUCHY,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecauchy @ V0s )
<=> ! [V1e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e )
=> ? [V2N: tyop_2Enum_2Enum] :
! [V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V3n @ V2N )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( V0s @ V3n ) @ ( V0s @ V2N ) ) ) @ V1e ) ) ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__CLOSED__IMP__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V0s )
& ( c_2Ereal__topology_2EClosed @ V0s ) )
=> ( c_2Ereal__topology_2Ecompact @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__LEMMA,axiom,
! [A_27a: $tType,V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ebounded__def @ V1s )
& ! [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0x @ V2n ) @ V1s ) )
=> ? [V3l: tyop_2Erealax_2Ereal,V4r: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V5m @ V6n )
=> ( c_2Eprim__rec_2E_3C @ ( V4r @ V5m ) @ ( V4r @ V6n ) ) )
& ! [V7e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V7e )
=> ? [V8N: tyop_2Enum_2Enum] :
! [V9n: tyop_2Enum_2Enum,V10i: A_27a] :
( ( c_2Earithmetic_2E_3C_3D @ V8N @ V9n )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V0x @ ( V4r @ V9n ) ) @ V3l ) ) @ V7e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__REAL__LEMMA,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0s @ V2n ) ) @ V1b )
=> ? [V3l: tyop_2Erealax_2Ereal,V4r: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V5m @ V6n )
=> ( c_2Eprim__rec_2E_3C @ ( V4r @ V5m ) @ ( V4r @ V6n ) ) )
& ! [V7e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V7e )
=> ? [V8N: tyop_2Enum_2Enum] :
! [V9n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V8N @ V9n )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V0s @ ( V4r @ V9n ) ) @ V3l ) ) @ V7e ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONVERGENT__BOUNDED__MONOTONE,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0s @ V2n ) ) @ V1b )
& ( ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V3m @ V4n )
=> ( c_2Ereal_2Ereal__lte @ ( V0s @ V3m ) @ ( V0s @ V4n ) ) )
| ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V5m @ V6n )
=> ( c_2Ereal_2Ereal__lte @ ( V0s @ V6n ) @ ( V0s @ V5m ) ) ) ) )
=> ? [V7l: tyop_2Erealax_2Ereal] :
! [V8e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V8e )
=> ? [V9N: tyop_2Enum_2Enum] :
! [V10n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V9N @ V10n )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V0s @ V10n ) @ V7l ) ) @ V8e ) ) ) ) ).
thf(thm_2Ereal__topology_2ECONVERGENT__BOUNDED__INCREASING,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V2m @ V3n )
=> ( c_2Ereal_2Ereal__lte @ ( V0s @ V2m ) @ ( V0s @ V3n ) ) )
& ! [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0s @ V4n ) ) @ V1b ) )
=> ? [V5l: tyop_2Erealax_2Ereal] :
! [V6e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V6e )
=> ? [V7N: tyop_2Enum_2Enum] :
! [V8n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V7N @ V8n )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V0s @ V8n ) @ V5l ) ) @ V6e ) ) ) ) ).
thf(thm_2Ereal__topology_2EMONOTONE__SUBSEQUENCE,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
? [V1r: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2m @ V3n )
=> ( c_2Eprim__rec_2E_3C @ ( V1r @ V2m ) @ ( V1r @ V3n ) ) )
& ( ! [V4m: tyop_2Enum_2Enum,V5n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V4m @ V5n )
=> ( c_2Ereal_2Ereal__lte @ ( V0s @ ( V1r @ V4m ) ) @ ( V0s @ ( V1r @ V5n ) ) ) )
| ! [V6m: tyop_2Enum_2Enum,V7n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V6m @ V7n )
=> ( c_2Ereal_2Ereal__lte @ ( V0s @ ( V1r @ V7n ) ) @ ( V0s @ ( V1r @ V6m ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELIM__SUBSEQUENCE,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1r: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V2l: tyop_2Erealax_2Ereal] :
( ( ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3m @ V4n )
=> ( c_2Eprim__rec_2E_3C @ ( V1r @ V3m ) @ ( V1r @ V4n ) ) )
& ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ V0s @ V2l @ c_2Ereal__topology_2Esequentially ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E @ tyop_2Enum_2Enum @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ V0s @ V1r ) @ V2l @ c_2Ereal__topology_2Esequentially ) ) ).
thf(thm_2Ereal__topology_2EMONOTONE__BIGGER,axiom,
! [V0r: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1m @ V2n )
=> ( c_2Eprim__rec_2E_3C @ ( V0r @ V1m ) @ ( V0r @ V2n ) ) )
=> ! [V3n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ V3n @ ( V0r @ V3n ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__AS__GDELTA,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
=> ( c_2Ereal__topology_2Egdelta @ V0s ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ? [V2u: tyop_2Erealax_2Ereal > $o,V3v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V2u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2u @ V3v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3v @ V0s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V2u )
& ( c_2Ereal__topology_2Ecompact @ V3v ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__ALT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ? [V2u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V2u )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V2u )
& ( c_2Ereal__topology_2Ecompact @ ( c_2Ereal__topology_2Eclosure @ V2u ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V2u ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__INTER__CBALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ? [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2e ) ) @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__INTER__CBALLS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
<=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ? [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ! [V3d: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V3d @ V2e )
=> ( c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V3d ) ) @ V0s ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
<=> ! [V1k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1k @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1k ) )
=> ? [V2u: tyop_2Erealax_2Ereal > $o,V3v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1k @ V2u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2u @ V3v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3v @ V0s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V2u )
& ( c_2Ereal__topology_2Ecompact @ V3v ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__COMPACT__ALT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
<=> ! [V1k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1k @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1k ) )
=> ? [V2u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1k @ V2u )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V2u )
& ( c_2Ereal__topology_2Ecompact @ ( c_2Ereal__topology_2Eclosure @ V2u ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eclosure @ V2u ) @ V0s ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__COMPACT__SUBOPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
<=> ! [V1k: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1k @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1k )
& ( c_2Ereal__topology_2EOpen @ V2t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1k @ V2t ) )
=> ? [V3u: tyop_2Erealax_2Ereal > $o,V4v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1k @ V3u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3u @ V4v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3u @ V2t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V4v @ V0s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V3u )
& ( c_2Ereal__topology_2Ecompact @ V4v ) ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IMP__LOCALLY__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V0s )
=> ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__IMP__LOCALLY__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
=> ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s ) ) ).
thf(thm_2Ereal__topology_2EIS__INTERVAL__IMP__LOCALLY__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Eis__interval @ V0s )
=> ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__UNIV,axiom,
c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__INTER,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V1t ) )
=> ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__OPEN__IN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t )
& ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V1t ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__CLOSED__IN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t )
& ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V1t ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__DELETE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
=> ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EDELETE @ tyop_2Erealax_2Ereal @ V0s @ V1a ) ) ) ).
thf(thm_2Ereal__topology_2EHOMEOMORPHIC__LOCAL__COMPACTNESS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ehomeomorphic @ V0s @ V1t )
=> ( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
= ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__TRANSLATION__EQ,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ V0a @ V2x )
@ V1s ) )
= ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V1s ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__CLOSED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2EClosed @ V0s )
= ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__OPEN__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V1t )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) @ V0s )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) @ V1t ) )
=> ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__CLOSED__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
& ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V1t )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) @ V0s )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) @ V1t ) )
=> ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2EOPEN__IN__LOCALLY__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V0s ) @ V1t )
<=> ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s )
& ! [V2k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V2k )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2k @ V0s ) )
=> ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2k ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V2k @ V1t ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__PROPER__IMAGE__EQ,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ! [V2k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2k @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
& ( c_2Ereal__topology_2Ecompact @ V2k ) )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2k ) ) ) ) ) ) )
=> ( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V1s )
= ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__PROPER__IMAGE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ! [V2k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V2k @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) )
& ( c_2Ereal__topology_2Ecompact @ V2k ) )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ V2k ) ) ) ) ) )
& ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V1s ) )
=> ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__topology_2EMUMFORD__LEMMA,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o,V2t: tyop_2Erealax_2Ereal > $o,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Econtinuous__on @ V0f @ V1s )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V1s ) @ V2t )
& ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V2t )
& ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V4x ) @ V3y ) ) ) ) ) )
=> ? [V5u: tyop_2Erealax_2Ereal > $o,V6v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V5u )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V2t ) @ V6v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V7x @ V1s ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V7x ) @ V3y ) ) ) )
@ V5u )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V6v )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EIMAGE @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0f @ V5u ) @ V6v )
& ! [V8k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V8k @ V6v )
& ( c_2Ereal__topology_2Ecompact @ V8k ) )
=> ( c_2Ereal__topology_2Ecompact
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V9x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V9x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9x @ V5u ) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( V0f @ V9x ) @ V8k ) ) ) ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__OPEN__INTER__CLOSURE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
=> ? [V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V1t )
& ( V0s
= ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1t @ ( c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__CLOSED__IN__OPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
=> ? [V1t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V1t )
& ( c_2Etopology_2Eclosed__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1t ) @ V0s ) ) ) ).
thf(thm_2Ereal__topology_2ELOCALLY__COMPACT__CLOSED__INTER__OPEN,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V0s )
<=> ? [V1t: tyop_2Erealax_2Ereal > $o,V2u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V1t )
& ( c_2Ereal__topology_2EOpen @ V2u )
& ( V0s
= ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V1t @ V2u ) ) ) ) ).
thf(thm_2Ereal__topology_2EBAIRE,axiom,
! [V0g: ( tyop_2Erealax_2Ereal > $o ) > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V1s )
& ( c_2Epred__set_2Ecountable @ ( tyop_2Erealax_2Ereal > $o ) @ V0g )
& ! [V2t: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2t @ V0g )
=> ( ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V2t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1s @ ( c_2Ereal__topology_2Eclosure @ V2t ) ) ) ) )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1s @ ( c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EBIGINTER @ tyop_2Erealax_2Ereal @ V0g ) ) ) ) ).
thf(thm_2Ereal__topology_2EBAIRE__ALT,axiom,
! [V0g: ( tyop_2Erealax_2Ereal > $o ) > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Epred__set_2Ecountable @ ( tyop_2Erealax_2Ereal > $o ) @ V0g )
& ( ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0g )
= V1s ) )
=> ? [V2t: tyop_2Erealax_2Ereal > $o,V3u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2t @ V0g )
& ( c_2Etopology_2Eopen__in @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Esubtopology @ tyop_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean @ V1s ) @ V3u )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V3u @ ( c_2Ereal__topology_2Eclosure @ V2t ) ) ) ) ).
thf(thm_2Ereal__topology_2ENOWHERE__DENSE__COUNTABLE__BIGUNION__CLOSED,axiom,
! [V0g: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2Ecountable @ ( tyop_2Erealax_2Ereal > $o ) @ V0g )
& ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0g )
=> ( ( c_2Ereal__topology_2EClosed @ V1s )
& ( ( c_2Ereal__topology_2Einterior @ V1s )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) )
=> ( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0g ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ENOWHERE__DENSE__COUNTABLE__BIGUNION,axiom,
! [V0g: ( tyop_2Erealax_2Ereal > $o ) > $o] :
( ( ( c_2Epred__set_2Ecountable @ ( tyop_2Erealax_2Ereal > $o ) @ V0g )
& ! [V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ V0g )
=> ( ( c_2Ereal__topology_2Einterior @ ( c_2Ereal__topology_2Eclosure @ V1s ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0g ) )
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ).
thf(thm_2Ereal__topology_2ESUBORDINATE__PARTITION__OF__UNITY,axiom,
! [V0c: ( tyop_2Erealax_2Ereal > $o ) > $o,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1s @ ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal @ V0c ) )
& ! [V2u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V2u @ V0c )
=> ( c_2Ereal__topology_2EOpen @ V2u ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1s )
=> ? [V4v: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V4v )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V4v )
& ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o )
@ ^ [V5u: tyop_2Erealax_2Ereal > $o] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ V5u @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V5u @ V0c ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V5u @ V4v ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ) ) ) ) )
=> ? [V6f: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ! [V7u: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V7u @ V0c )
=> ( ( c_2Ereal__topology_2Econtinuous__on @ ( V6f @ V7u ) @ V1s )
& ! [V8x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V8x @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V6f @ V7u @ V8x ) ) ) ) )
& ! [V9x: tyop_2Erealax_2Ereal,V10u: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V10u @ V0c )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9x @ V1s )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V9x @ V10u ) ) )
=> ( ( V6f @ V10u @ V9x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ! [V11x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V11x @ V1s )
=> ( ( c_2Eiterate_2ESum @ ( tyop_2Erealax_2Ereal > $o ) @ V0c
@ ^ [V12u: tyop_2Erealax_2Ereal > $o] : ( V6f @ V12u @ V11x ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V13x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V13x @ V1s )
=> ? [V14n: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EOpen @ V14n )
& ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V13x @ V14n )
& ( c_2Epred__set_2EFINITE @ ( tyop_2Erealax_2Ereal > $o )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ ( tyop_2Erealax_2Ereal > $o )
@ ^ [V15u: tyop_2Erealax_2Ereal > $o] :
( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ V15u
@ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( tyop_2Erealax_2Ereal > $o ) @ V15u @ V0c )
@ ( c_2Ebool_2E_7E
@ ( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V16x: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V16x @ V14n ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V6f @ V15u @ V16x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------