ITP001 Axioms: ITP149^5.ax
%------------------------------------------------------------------------------
% File : ITP149^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : real_topology^2.ax [Gau20]
% : HL4149^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 1920 ( 104 unt; 83 typ; 0 def)
% Number of atoms : 69325 (1196 equ; 0 cnn)
% Maximal formula atoms : 244 ( 36 avg)
% Number of connectives : 91227 ( 384 ~; 105 |;2214 &;83194 @)
% ( 557 <=>;4773 =>; 0 <=; 0 <~>)
% Maximal formula depth : 47 ( 15 avg;83194 nst)
% Number of types : 5 ( 4 usr)
% Number of type conns : 25 ( 25 >; 0 *; 0 +; 0 <<)
% Number of symbols : 183 ( 181 usr; 158 con; 0-2 aty)
% Number of variables : 6688 ( 756 ^5494 !; 438 ?;6688 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Ereal__topology_2Enet,type,
ty_2Ereal__topology_2Enet: del > del ).
thf(tp_c_2Ereal__topology_2E_2D_2D_3E,type,
c_2Ereal__topology_2E_2D_2D_3E: del > $i ).
thf(mem_c_2Ereal__topology_2E_2D_2D_3E,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ( ty_2Ereal__topology_2Enet @ A_27a ) @ bool ) ) ) ) ).
thf(stp_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,
tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal: $tType ).
thf(stp_inj_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,
inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal > $i ).
thf(stp_surj_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,
surj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal: $i > tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal ).
thf(stp_inj_surj_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ( mem @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X ) @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ).
thf(stp_iso_mem_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ( X
= ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( surj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X ) ) ) ) ).
thf(tp_c_2Ereal__topology_2ECLOSED__interval,type,
c_2Ereal__topology_2ECLOSED__interval: $i ).
thf(mem_c_2Ereal__topology_2ECLOSED__interval,axiom,
mem @ c_2Ereal__topology_2ECLOSED__interval @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ereal__topology_2EClosed,type,
c_2Ereal__topology_2EClosed: $i ).
thf(mem_c_2Ereal__topology_2EClosed,axiom,
mem @ c_2Ereal__topology_2EClosed @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ereal__topology_2EDist,type,
c_2Ereal__topology_2EDist: $i ).
thf(mem_c_2Ereal__topology_2EDist,axiom,
mem @ c_2Ereal__topology_2EDist @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Ereal__topology_2EDist,type,
fo__c_2Ereal__topology_2EDist: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Ereal__topology_2EDist,axiom,
! [X0: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Ereal__topology_2EDist @ X0 ) )
= ( ap @ c_2Ereal__topology_2EDist @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Ereal__topology_2EOPEN__interval,type,
c_2Ereal__topology_2EOPEN__interval: $i ).
thf(mem_c_2Ereal__topology_2EOPEN__interval,axiom,
mem @ c_2Ereal__topology_2EOPEN__interval @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ereal__topology_2EOpen,type,
c_2Ereal__topology_2EOpen: $i ).
thf(mem_c_2Ereal__topology_2EOpen,axiom,
mem @ c_2Ereal__topology_2EOpen @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(stp_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,type,
tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal: $tType ).
thf(stp_inj_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,type,
inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal: tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal > $i ).
thf(stp_surj_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,type,
surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal: $i > tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal ).
thf(stp_inj_surj_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ ( inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal] : ( mem @ ( inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ X ) @ ( ty_2Ereal__topology_2Enet @ ty_2Erealax_2Ereal ) ) ).
thf(stp_iso_mem_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Ereal__topology_2Enet @ ty_2Erealax_2Ereal ) )
=> ( X
= ( inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ X ) ) ) ) ).
thf(tp_c_2Ereal__topology_2Eat,type,
c_2Ereal__topology_2Eat: $i ).
thf(mem_c_2Ereal__topology_2Eat,axiom,
mem @ c_2Ereal__topology_2Eat @ ( arr @ ty_2Erealax_2Ereal @ ( ty_2Ereal__topology_2Enet @ ty_2Erealax_2Ereal ) ) ).
thf(tp_c_2Ereal__topology_2Eat__infinity,type,
c_2Ereal__topology_2Eat__infinity: $i ).
thf(mem_c_2Ereal__topology_2Eat__infinity,axiom,
mem @ c_2Ereal__topology_2Eat__infinity @ ( ty_2Ereal__topology_2Enet @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Ereal__topology_2Eat__neginfinity,type,
c_2Ereal__topology_2Eat__neginfinity: $i ).
thf(mem_c_2Ereal__topology_2Eat__neginfinity,axiom,
mem @ c_2Ereal__topology_2Eat__neginfinity @ ( ty_2Ereal__topology_2Enet @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Ereal__topology_2Eat__posinfinity,type,
c_2Ereal__topology_2Eat__posinfinity: $i ).
thf(mem_c_2Ereal__topology_2Eat__posinfinity,axiom,
mem @ c_2Ereal__topology_2Eat__posinfinity @ ( ty_2Ereal__topology_2Enet @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Ereal__topology_2Eball,type,
c_2Ereal__topology_2Eball: $i ).
thf(mem_c_2Ereal__topology_2Eball,axiom,
mem @ c_2Ereal__topology_2Eball @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Ebetween,type,
c_2Ereal__topology_2Ebetween: $i ).
thf(mem_c_2Ereal__topology_2Ebetween,axiom,
mem @ c_2Ereal__topology_2Ebetween @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Ebilinear,type,
c_2Ereal__topology_2Ebilinear: $i ).
thf(mem_c_2Ereal__topology_2Ebilinear,axiom,
mem @ c_2Ereal__topology_2Ebilinear @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Ebounded__def,type,
c_2Ereal__topology_2Ebounded__def: $i ).
thf(mem_c_2Ereal__topology_2Ebounded__def,axiom,
mem @ c_2Ereal__topology_2Ebounded__def @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Ecauchy,type,
c_2Ereal__topology_2Ecauchy: $i ).
thf(mem_c_2Ereal__topology_2Ecauchy,axiom,
mem @ c_2Ereal__topology_2Ecauchy @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Ecball,type,
c_2Ereal__topology_2Ecball: $i ).
thf(mem_c_2Ereal__topology_2Ecball,axiom,
mem @ c_2Ereal__topology_2Ecball @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Eclosed__segment,type,
c_2Ereal__topology_2Eclosed__segment: $i ).
thf(mem_c_2Ereal__topology_2Eclosed__segment,axiom,
mem @ c_2Ereal__topology_2Eclosed__segment @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Eclosest__point,type,
c_2Ereal__topology_2Eclosest__point: $i ).
thf(mem_c_2Ereal__topology_2Eclosest__point,axiom,
mem @ c_2Ereal__topology_2Eclosest__point @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ).
thf(tp_c_2Ereal__topology_2Eclosure,type,
c_2Ereal__topology_2Eclosure: $i ).
thf(mem_c_2Ereal__topology_2Eclosure,axiom,
mem @ c_2Ereal__topology_2Eclosure @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Ecollinear,type,
c_2Ereal__topology_2Ecollinear: $i ).
thf(mem_c_2Ereal__topology_2Ecollinear,axiom,
mem @ c_2Ereal__topology_2Ecollinear @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Ecompact,type,
c_2Ereal__topology_2Ecompact: $i ).
thf(mem_c_2Ereal__topology_2Ecompact,axiom,
mem @ c_2Ereal__topology_2Ecompact @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Ecomplete,type,
c_2Ereal__topology_2Ecomplete: $i ).
thf(mem_c_2Ereal__topology_2Ecomplete,axiom,
mem @ c_2Ereal__topology_2Ecomplete @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Ecomponents,type,
c_2Ereal__topology_2Ecomponents: $i ).
thf(mem_c_2Ereal__topology_2Ecomponents,axiom,
mem @ c_2Ereal__topology_2Ecomponents @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Econdensation__point__of,type,
c_2Ereal__topology_2Econdensation__point__of: $i ).
thf(mem_c_2Ereal__topology_2Econdensation__point__of,axiom,
mem @ c_2Ereal__topology_2Econdensation__point__of @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Econnected,type,
c_2Ereal__topology_2Econnected: $i ).
thf(mem_c_2Ereal__topology_2Econnected,axiom,
mem @ c_2Ereal__topology_2Econnected @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Econnected__component,type,
c_2Ereal__topology_2Econnected__component: $i ).
thf(mem_c_2Ereal__topology_2Econnected__component,axiom,
mem @ c_2Ereal__topology_2Econnected__component @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ) ).
thf(tp_c_2Ereal__topology_2Econtinuous,type,
c_2Ereal__topology_2Econtinuous: del > $i ).
thf(mem_c_2Ereal__topology_2Econtinuous,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) @ ( arr @ ( ty_2Ereal__topology_2Enet @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Ereal__topology_2Econtinuous__on,type,
c_2Ereal__topology_2Econtinuous__on: $i ).
thf(mem_c_2Ereal__topology_2Econtinuous__on,axiom,
mem @ c_2Ereal__topology_2Econtinuous__on @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Edependent,type,
c_2Ereal__topology_2Edependent: $i ).
thf(mem_c_2Ereal__topology_2Edependent,axiom,
mem @ c_2Ereal__topology_2Edependent @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Ediameter,type,
c_2Ereal__topology_2Ediameter: $i ).
thf(mem_c_2Ereal__topology_2Ediameter,axiom,
mem @ c_2Ereal__topology_2Ediameter @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Ereal__topology_2Edim,type,
c_2Ereal__topology_2Edim: $i ).
thf(mem_c_2Ereal__topology_2Edim,axiom,
mem @ c_2Ereal__topology_2Edim @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ty_2Enum_2Enum ) ).
thf(tp_c_2Ereal__topology_2Eeuclidean,type,
c_2Ereal__topology_2Eeuclidean: $i ).
thf(mem_c_2Ereal__topology_2Eeuclidean,axiom,
mem @ c_2Ereal__topology_2Eeuclidean @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Ereal__topology_2Eeventually,type,
c_2Ereal__topology_2Eeventually: del > $i ).
thf(mem_c_2Ereal__topology_2Eeventually,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Ereal__topology_2Enet @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Ereal__topology_2Efrom,type,
c_2Ereal__topology_2Efrom: $i ).
thf(mem_c_2Ereal__topology_2Efrom,axiom,
mem @ c_2Ereal__topology_2Efrom @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Efrontier,type,
c_2Ereal__topology_2Efrontier: $i ).
thf(mem_c_2Ereal__topology_2Efrontier,axiom,
mem @ c_2Ereal__topology_2Efrontier @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Efsigma,type,
c_2Ereal__topology_2Efsigma: $i ).
thf(mem_c_2Ereal__topology_2Efsigma,axiom,
mem @ c_2Ereal__topology_2Efsigma @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Egdelta,type,
c_2Ereal__topology_2Egdelta: $i ).
thf(mem_c_2Ereal__topology_2Egdelta,axiom,
mem @ c_2Ereal__topology_2Egdelta @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Ehausdist,type,
c_2Ereal__topology_2Ehausdist: $i ).
thf(mem_c_2Ereal__topology_2Ehausdist,axiom,
mem @ c_2Ereal__topology_2Ehausdist @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Ereal__topology_2Ehomeomorphic,type,
c_2Ereal__topology_2Ehomeomorphic: $i ).
thf(mem_c_2Ereal__topology_2Ehomeomorphic,axiom,
mem @ c_2Ereal__topology_2Ehomeomorphic @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Ehomeomorphism,type,
c_2Ereal__topology_2Ehomeomorphism: $i ).
thf(mem_c_2Ereal__topology_2Ehomeomorphism,axiom,
mem @ c_2Ereal__topology_2Ehomeomorphism @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Ein__direction,type,
c_2Ereal__topology_2Ein__direction: $i ).
thf(mem_c_2Ereal__topology_2Ein__direction,axiom,
mem @ c_2Ereal__topology_2Ein__direction @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ( ty_2Ereal__topology_2Enet @ ty_2Erealax_2Ereal ) ) ) ).
thf(tp_c_2Ereal__topology_2Eindependent,type,
c_2Ereal__topology_2Eindependent: $i ).
thf(mem_c_2Ereal__topology_2Eindependent,axiom,
mem @ c_2Ereal__topology_2Eindependent @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Einfsum,type,
c_2Ereal__topology_2Einfsum: $i ).
thf(mem_c_2Ereal__topology_2Einfsum,axiom,
mem @ c_2Ereal__topology_2Einfsum @ ( arr @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ) ).
thf(tp_c_2Ereal__topology_2Einterior,type,
c_2Ereal__topology_2Einterior: $i ).
thf(mem_c_2Ereal__topology_2Einterior,axiom,
mem @ c_2Ereal__topology_2Einterior @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Eis__interval,type,
c_2Ereal__topology_2Eis__interval: $i ).
thf(mem_c_2Ereal__topology_2Eis__interval,axiom,
mem @ c_2Ereal__topology_2Eis__interval @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Eisnet,type,
c_2Ereal__topology_2Eisnet: del > del > $i ).
thf(mem_c_2Ereal__topology_2Eisnet,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ereal__topology_2Eisnet @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Elim__def,type,
c_2Ereal__topology_2Elim__def: del > $i ).
thf(mem_c_2Ereal__topology_2Elim__def,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__topology_2Elim__def @ A_27a ) @ ( arr @ ( ty_2Ereal__topology_2Enet @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ) ) ).
thf(tp_c_2Ereal__topology_2Elimit__point__of,type,
c_2Ereal__topology_2Elimit__point__of: $i ).
thf(mem_c_2Ereal__topology_2Elimit__point__of,axiom,
mem @ c_2Ereal__topology_2Elimit__point__of @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Elinear,type,
c_2Ereal__topology_2Elinear: $i ).
thf(mem_c_2Ereal__topology_2Elinear,axiom,
mem @ c_2Ereal__topology_2Elinear @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Elocally,type,
c_2Ereal__topology_2Elocally: $i ).
thf(mem_c_2Ereal__topology_2Elocally,axiom,
mem @ c_2Ereal__topology_2Elocally @ ( arr @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Emidpoint,type,
c_2Ereal__topology_2Emidpoint: $i ).
thf(mem_c_2Ereal__topology_2Emidpoint,axiom,
mem @ c_2Ereal__topology_2Emidpoint @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Ereal__topology_2Emidpoint,type,
fo__c_2Ereal__topology_2Emidpoint: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Ereal__topology_2Emidpoint,axiom,
! [X0: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Ereal__topology_2Emidpoint @ X0 ) )
= ( ap @ c_2Ereal__topology_2Emidpoint @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Ereal__topology_2Emk__net,type,
c_2Ereal__topology_2Emk__net: del > $i ).
thf(mem_c_2Ereal__topology_2Emk__net,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__topology_2Emk__net @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( ty_2Ereal__topology_2Enet @ A_27a ) ) ) ).
thf(tp_c_2Ereal__topology_2Enetlimit,type,
c_2Ereal__topology_2Enetlimit: del > $i ).
thf(mem_c_2Ereal__topology_2Enetlimit,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__topology_2Enetlimit @ A_27a ) @ ( arr @ ( ty_2Ereal__topology_2Enet @ A_27a ) @ A_27a ) ) ).
thf(tp_c_2Ereal__topology_2Enetord,type,
c_2Ereal__topology_2Enetord: del > $i ).
thf(mem_c_2Ereal__topology_2Enetord,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ ( arr @ ( ty_2Ereal__topology_2Enet @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Ereal__topology_2Eopen__segment,type,
c_2Ereal__topology_2Eopen__segment: $i ).
thf(mem_c_2Ereal__topology_2Eopen__segment,axiom,
mem @ c_2Ereal__topology_2Eopen__segment @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Epairwise,type,
c_2Ereal__topology_2Epairwise: del > $i ).
thf(mem_c_2Ereal__topology_2Epairwise,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__topology_2Epairwise @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Ereal__topology_2Epermutes,type,
c_2Ereal__topology_2Epermutes: del > $i ).
thf(mem_c_2Ereal__topology_2Epermutes,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__topology_2Epermutes @ A_27a ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(stp_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,type,
tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum: $tType ).
thf(stp_inj_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,type,
inj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum: tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum > $i ).
thf(stp_surj_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,type,
surj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum: $i > tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum ).
thf(stp_inj_surj_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,axiom,
! [X: tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum] :
( ( surj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ ( inj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,axiom,
! [X: tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum] : ( mem @ ( inj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ X ) @ ( ty_2Ereal__topology_2Enet @ ty_2Enum_2Enum ) ) ).
thf(stp_iso_mem_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Ereal__topology_2Enet @ ty_2Enum_2Enum ) )
=> ( X
= ( inj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ ( surj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ X ) ) ) ) ).
thf(tp_c_2Ereal__topology_2Esequentially,type,
c_2Ereal__topology_2Esequentially: $i ).
thf(mem_c_2Ereal__topology_2Esequentially,axiom,
mem @ c_2Ereal__topology_2Esequentially @ ( ty_2Ereal__topology_2Enet @ ty_2Enum_2Enum ) ).
thf(tp_c_2Ereal__topology_2Esetdist,type,
c_2Ereal__topology_2Esetdist: $i ).
thf(mem_c_2Ereal__topology_2Esetdist,axiom,
mem @ c_2Ereal__topology_2Esetdist @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Ereal__topology_2Espan,type,
c_2Ereal__topology_2Espan: $i ).
thf(mem_c_2Ereal__topology_2Espan,axiom,
mem @ c_2Ereal__topology_2Espan @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Esphere,type,
c_2Ereal__topology_2Esphere: $i ).
thf(mem_c_2Ereal__topology_2Esphere,axiom,
mem @ c_2Ereal__topology_2Esphere @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Esubspace,type,
c_2Ereal__topology_2Esubspace: $i ).
thf(mem_c_2Ereal__topology_2Esubspace,axiom,
mem @ c_2Ereal__topology_2Esubspace @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ).
thf(tp_c_2Ereal__topology_2Esubtopology,type,
c_2Ereal__topology_2Esubtopology: del > $i ).
thf(mem_c_2Ereal__topology_2Esubtopology,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ ( arr @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( ty_2Etopology_2Etopology @ A_27a ) ) ) ) ).
thf(tp_c_2Ereal__topology_2Esummable,type,
c_2Ereal__topology_2Esummable: $i ).
thf(mem_c_2Ereal__topology_2Esummable,axiom,
mem @ c_2Ereal__topology_2Esummable @ ( arr @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Esums,type,
c_2Ereal__topology_2Esums: $i ).
thf(mem_c_2Ereal__topology_2Esums,axiom,
mem @ c_2Ereal__topology_2Esums @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ( arr @ ty_2Enum_2Enum @ bool ) @ bool ) ) ) ).
thf(tp_c_2Ereal__topology_2Etrivial__limit,type,
c_2Ereal__topology_2Etrivial__limit: del > $i ).
thf(mem_c_2Ereal__topology_2Etrivial__limit,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ ( arr @ ( ty_2Ereal__topology_2Enet @ A_27a ) @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Euniformly__continuous__on,type,
c_2Ereal__topology_2Euniformly__continuous__on: $i ).
thf(mem_c_2Ereal__topology_2Euniformly__continuous__on,axiom,
mem @ c_2Ereal__topology_2Euniformly__continuous__on @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) ) ).
thf(tp_c_2Ereal__topology_2Ewithin,type,
c_2Ereal__topology_2Ewithin: del > $i ).
thf(mem_c_2Ereal__topology_2Ewithin,axiom,
! [A_27a: del] : ( mem @ ( c_2Ereal__topology_2Ewithin @ A_27a ) @ ( arr @ ( ty_2Ereal__topology_2Enet @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( ty_2Ereal__topology_2Enet @ A_27a ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LE__SQUARE__ABS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__EQ__SQUARE__ABS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__HALF,axiom,
( ! [V0e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) ) )
& ! [V1e: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
= V1e )
& ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
= V2e ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__SUBSET__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V2t ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) ) )
<=> ? [V3s_27: $i] :
( ( mem @ V3s_27 @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3s_27 ) @ V1s ) )
& ( V2t
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V3s_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEXISTS__FINITE__SUBSET__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ? [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3t ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V1f ) @ V2s ) ) )
& ( p @ ( ap @ V0P @ V3t ) ) )
<=> ? [V4t: $i] :
( ( mem @ V4t @ ( arr @ A_27b @ bool ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V4t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V4t ) @ V2s ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V1f ) @ V4t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFORALL__FINITE__SUBSET__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3t ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V1f ) @ V2s ) ) ) )
=> ( p @ ( ap @ V0P @ V3t ) ) ) )
<=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V4t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V4t ) @ V2s ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V1f ) @ V4t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFORALL__IN__GSPEC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,V0Q: $i] :
( ( mem @ V0Q @ ( arr @ A_27b @ bool ) )
=> ( ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V3z: $i] :
( ( mem @ V3z @ A_27b )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3z )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ V2f @ V4x ) ) @ ( ap @ V1P @ V4x ) ) ) ) ) )
=> ( p @ ( ap @ V0Q @ V3z ) ) ) )
<=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ V1P @ V5x ) )
=> ( p @ ( ap @ V0Q @ ( ap @ V2f @ V5x ) ) ) ) ) ) ) )
& ! [V6P: $i] :
( ( mem @ V6P @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27c @ ( arr @ A_27d @ A_27b ) ) )
=> ( ! [V8z: $i] :
( ( mem @ V8z @ A_27b )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V8z )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ ( ty_2Epair_2Eprod @ A_27b @ bool ) )
@ ( lam @ A_27c
@ ^ [V9x: $i] :
( lam @ A_27d
@ ^ [V10y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ ( ap @ V7f @ V9x ) @ V10y ) ) @ ( ap @ ( ap @ V6P @ V9x ) @ V10y ) ) ) ) ) ) ) )
=> ( p @ ( ap @ V0Q @ V8z ) ) ) )
<=> ! [V11x: $i] :
( ( mem @ V11x @ A_27c )
=> ! [V12y: $i] :
( ( mem @ V12y @ A_27d )
=> ( ( p @ ( ap @ ( ap @ V6P @ V11x ) @ V12y ) )
=> ( p @ ( ap @ V0Q @ ( ap @ ( ap @ V7f @ V11x ) @ V12y ) ) ) ) ) ) ) ) )
& ! [V13P: $i] :
( ( mem @ V13P @ ( arr @ A_27e @ ( arr @ A_27f @ ( arr @ A_27g @ bool ) ) ) )
=> ! [V14f: $i] :
( ( mem @ V14f @ ( arr @ A_27e @ ( arr @ A_27f @ ( arr @ A_27g @ A_27b ) ) ) )
=> ( ! [V15z: $i] :
( ( mem @ V15z @ A_27b )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V15z )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ ( ty_2Epair_2Eprod @ A_27e @ ( ty_2Epair_2Eprod @ A_27f @ A_27g ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27e @ ( ty_2Epair_2Eprod @ A_27f @ A_27g ) @ ( ty_2Epair_2Eprod @ A_27b @ bool ) )
@ ( lam @ A_27e
@ ^ [V16w: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27f @ A_27g @ ( ty_2Epair_2Eprod @ A_27b @ bool ) )
@ ( lam @ A_27f
@ ^ [V17x: $i] :
( lam @ A_27g
@ ^ [V18y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ ( ap @ ( ap @ V14f @ V16w ) @ V17x ) @ V18y ) ) @ ( ap @ ( ap @ ( ap @ V13P @ V16w ) @ V17x ) @ V18y ) ) ) ) ) ) ) ) ) )
=> ( p @ ( ap @ V0Q @ V15z ) ) ) )
<=> ! [V19w: $i] :
( ( mem @ V19w @ A_27e )
=> ! [V20x: $i] :
( ( mem @ V20x @ A_27f )
=> ! [V21y: $i] :
( ( mem @ V21y @ A_27g )
=> ( ( p @ ( ap @ ( ap @ ( ap @ V13P @ V19w ) @ V20x ) @ V21y ) )
=> ( p @ ( ap @ V0Q @ ( ap @ ( ap @ ( ap @ V14f @ V19w ) @ V20x ) @ V21y ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEXISTS__IN__GSPEC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,V0Q: $i] :
( ( mem @ V0Q @ ( arr @ A_27b @ bool ) )
=> ( ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ? [V3z: $i] :
( ( mem @ V3z @ A_27b )
& ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3z )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ V2f @ V4x ) ) @ ( ap @ V1P @ V4x ) ) ) ) ) )
& ( p @ ( ap @ V0Q @ V3z ) ) )
<=> ? [V5x: $i] :
( ( mem @ V5x @ A_27a )
& ( p @ ( ap @ V1P @ V5x ) )
& ( p @ ( ap @ V0Q @ ( ap @ V2f @ V5x ) ) ) ) ) ) )
& ! [V6P: $i] :
( ( mem @ V6P @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27c @ ( arr @ A_27d @ A_27b ) ) )
=> ( ? [V8z: $i] :
( ( mem @ V8z @ A_27b )
& ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V8z )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ ( ty_2Epair_2Eprod @ A_27b @ bool ) )
@ ( lam @ A_27c
@ ^ [V9x: $i] :
( lam @ A_27d
@ ^ [V10y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ ( ap @ V7f @ V9x ) @ V10y ) ) @ ( ap @ ( ap @ V6P @ V9x ) @ V10y ) ) ) ) ) ) ) )
& ( p @ ( ap @ V0Q @ V8z ) ) )
<=> ? [V11x: $i] :
( ( mem @ V11x @ A_27c )
& ? [V12y: $i] :
( ( mem @ V12y @ A_27d )
& ( p @ ( ap @ ( ap @ V6P @ V11x ) @ V12y ) )
& ( p @ ( ap @ V0Q @ ( ap @ ( ap @ V7f @ V11x ) @ V12y ) ) ) ) ) ) ) )
& ! [V13P: $i] :
( ( mem @ V13P @ ( arr @ A_27e @ ( arr @ A_27f @ ( arr @ A_27g @ bool ) ) ) )
=> ! [V14f: $i] :
( ( mem @ V14f @ ( arr @ A_27e @ ( arr @ A_27f @ ( arr @ A_27g @ A_27b ) ) ) )
=> ( ? [V15z: $i] :
( ( mem @ V15z @ A_27b )
& ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V15z )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ ( ty_2Epair_2Eprod @ A_27e @ ( ty_2Epair_2Eprod @ A_27f @ A_27g ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27e @ ( ty_2Epair_2Eprod @ A_27f @ A_27g ) @ ( ty_2Epair_2Eprod @ A_27b @ bool ) )
@ ( lam @ A_27e
@ ^ [V16w: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27f @ A_27g @ ( ty_2Epair_2Eprod @ A_27b @ bool ) )
@ ( lam @ A_27f
@ ^ [V17x: $i] :
( lam @ A_27g
@ ^ [V18y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ ( ap @ ( ap @ V14f @ V16w ) @ V17x ) @ V18y ) ) @ ( ap @ ( ap @ ( ap @ V13P @ V16w ) @ V17x ) @ V18y ) ) ) ) ) ) ) ) ) )
& ( p @ ( ap @ V0Q @ V15z ) ) )
<=> ? [V19w: $i] :
( ( mem @ V19w @ A_27e )
& ? [V20x: $i] :
( ( mem @ V20x @ A_27f )
& ? [V21y: $i] :
( ( mem @ V21y @ A_27g )
& ( p @ ( ap @ ( ap @ ( ap @ V13P @ V19w ) @ V20x ) @ V21y ) )
& ( p @ ( ap @ V0Q @ ( ap @ ( ap @ ( ap @ V14f @ V19w ) @ V20x ) @ V21y ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEMPTY__BIGUNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1t ) @ V0s ) )
=> ( V1t
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LT__RNEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUPPER__BOUND__FINITE__SET,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ? [V2a: tp__ty_2Enum_2Enum] :
! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__BOUNDS__LT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1k: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1k ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1k ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1k ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBIGUNION__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( arr @ A_27b @ bool ) ) @ V0f ) @ V1s ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V2y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V2y )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2y ) @ ( ap @ V0f @ V3x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBIGINTER__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( arr @ A_27b @ bool ) ) @ V0f ) @ V1s ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V2y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V2y )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2y ) @ ( ap @ V0f @ V3x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LE__LMUL1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELE__EXISTS,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ? [V2d: tp__ty_2Enum_2Enum] :
( V1n
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V2d ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDS__LINEAR,axiom,
! [V0A: tp__ty_2Enum_2Enum,V1B: tp__ty_2Enum_2Enum,V2C: tp__ty_2Enum_2Enum] :
( ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0A ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1B ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( inj__ty_2Enum_2Enum @ V2C ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0A ) ) @ ( inj__ty_2Enum_2Enum @ V1B ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDS__LINEAR__0,axiom,
! [V0A: tp__ty_2Enum_2Enum,V1B: tp__ty_2Enum_2Enum] :
( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0A ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( inj__ty_2Enum_2Enum @ V1B ) ) )
<=> ( V0A = fo__c_2Enum_2E0 ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LE__BETWEEN,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EWLOG__LE,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ( ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
<=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
=> ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__POWERSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1t: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V1t ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELE__ADD,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELE__ADDR,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EADD__SUB2,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
= V1n ) ).
thf(conj_thm_2Ereal__topology_2EADD__SUBR2,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= fo__c_2Enum_2E0 ) ).
thf(conj_thm_2Ereal__topology_2EADD__SUBR,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= fo__c_2Enum_2E0 ) ).
thf(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LE__EQ,axiom,
! [V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ( ! [V1x: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) )
& ! [V2x: tp__ty_2Enum_2Enum,V3y: tp__ty_2Enum_2Enum,V4z: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V3y ) ) )
& ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V3y ) ) @ ( inj__ty_2Enum_2Enum @ V4z ) ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V4z ) ) ) ) )
=> ( ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) )
<=> ! [V7n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V7n ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LE,axiom,
! [V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ( ! [V1x: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) )
& ! [V2x: tp__ty_2Enum_2Enum,V3y: tp__ty_2Enum_2Enum,V4z: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V3y ) ) )
& ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V3y ) ) @ ( inj__ty_2Enum_2Enum @ V4z ) ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V4z ) ) ) )
& ! [V5n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) )
=> ! [V6m: tp__ty_2Enum_2Enum,V7n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ ( inj__ty_2Enum_2Enum @ V7n ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELAMBDA__PAIR,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27b
@ ^ [V2y: $i] : ( ap @ ( ap @ V0P @ V1x ) @ V2y ) ) ) )
= ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V3p: $i] : ( ap @ ( ap @ V0P @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V3p ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V3p ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENOT__EQ,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27a )
=> ( ( V0a != V1b )
<=> ( V0a != V1b ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EABS__LE__0,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__OF__NUM__GE,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPOWERSET__CLAUSES,axiom,
! [A_27a: del,A_27b: del] :
( ( ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27b @ bool ) @ ( arr @ A_27b @ bool ) )
@ ( lam @ ( arr @ A_27b @ bool )
@ ^ [V0s: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ bool ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27b @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27b @ bool ) ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V3s: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V3s ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1a ) @ V2t ) ) ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EUNION @ ( arr @ A_27a @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V4s: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V4s ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V4s ) @ V2t ) ) ) ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V5s: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1a ) @ V5s ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V6s: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V6s ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V6s ) @ V2t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LT__LCANCEL__IMP,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESIMPLE__IMAGE__GEN,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ V0f @ V2x ) ) @ ( ap @ V1P @ V2x ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ V1P @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSET__BIGUNION,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V0f ) @ V1g ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0f ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LT__POW2,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFUN__IN__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSET__ANTISYM__EQ,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V0s ) ) )
<=> ( V0s = V1t ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIFF__BIGINTER,axiom,
! [A_27a: del,V0u: $i] :
( ( mem @ V0u @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0u ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1s ) )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2t: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0u ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBIGINTER__BIGUNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
@ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1t: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1t ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBIGUNION__BIGINTER,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
@ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1t: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1t ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__POW__1__LE,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__POW__LE__1,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LT__INV2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__INV__1__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUM__GP__BASIC,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2i: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUM__GP__MULTIPLIED,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V3i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUM__GP,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V3i ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ) ).
thf(stp_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,type,
tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o: $tType ).
thf(stp_inj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,type,
inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o > $i ).
thf(stp_surj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,type,
surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o: $i > tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o ).
thf(stp_inj_surj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,axiom,
! [X: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,axiom,
! [X: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o] : ( mem @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o @ X ) @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(stp_iso_mem_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
=> ( X
= ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o @ ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o @ X ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__SYM,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4y: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V4y ) @ V5x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4y ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V0s ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUM__ABS__TRIANGLE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p
@ ( ap
@ ( ap @ c_2Ereal_2Ereal__lte
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s )
@ ( lam @ A_27a
@ ^ [V3a: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V1f @ V3a ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ V1f ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIMAGE__SING,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ ( ap @ V0f @ V1a ) ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__WLOG__LE,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ( ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
<=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) )
=> ! [V5x: tp__ty_2Erealax_2Ereal,V6y: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Epairwise,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epairwise @ A_27a ) @ V0r ) @ V1s ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) )
& ( V2x != V3y ) )
=> ( p @ ( ap @ ( ap @ V0r @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPAIRWISE__EMPTY,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epairwise @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> $true ) ) ).
thf(conj_thm_2Ereal__topology_2EPAIRWISE__SING,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epairwise @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
<=> $true ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPAIRWISE__MONO,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epairwise @ A_27a ) @ V0r ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epairwise @ A_27a ) @ V0r ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPAIRWISE__INSERT,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epairwise @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ V2s ) ) )
<=> ( ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V2s ) )
& ( V3y != V1x ) )
=> ( ( p @ ( ap @ ( ap @ V0r @ V1x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ V0r @ V3y ) @ V1x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epairwise @ A_27a ) @ V0r ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPAIRWISE__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27b @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epairwise @ A_27a ) @ V1r ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V2f ) @ V0s ) ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Epairwise @ A_27b )
@ ( lam @ A_27b
@ ^ [V3x: $i] :
( lam @ A_27b
@ ^ [V4y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V2f @ V3x ) ) @ ( ap @ V2f @ V4y ) ) ) ) @ ( ap @ ( ap @ V1r @ ( ap @ V2f @ V3x ) ) @ ( ap @ V2f @ V4y ) ) ) ) ) )
@ V0s ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Epermutes,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epermutes @ A_27a ) @ V0p ) @ V1s ) )
<=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( ( ap @ V0p @ V2x )
= V2x ) ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V0p @ V4x ) ) @ V3y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPERMUTES__IMAGE,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epermutes @ A_27a ) @ V0p ) @ V1s ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V0p ) @ V1s )
= V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPERMUTES__INJECTIVE,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epermutes @ A_27a ) @ V0p ) @ V1s ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( ap @ V0p @ V2x )
= ( ap @ V0p @ V3y ) )
<=> ( V2x = V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEXISTS__IN__INSERT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1a ) @ V2s ) ) )
& ( p @ ( ap @ V0P @ V3x ) ) )
<=> ( ( p @ ( ap @ V0P @ V1a ) )
| ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) )
& ( p @ ( ap @ V0P @ V4x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDEPENDENT__CHOICE__FIXED,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V2a ) )
& ! [V3n: tp__ty_2Enum_2Enum,V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V4x ) )
=> ? [V5y: $i] :
( ( mem @ V5y @ A_27a )
& ( p @ ( ap @ ( ap @ V0P @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V5y ) )
& ( p @ ( ap @ ( ap @ ( ap @ V1R @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V4x ) @ V5y ) ) ) ) ) )
=> ? [V6f: $i] :
( ( mem @ V6f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( ( ap @ V6f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= V2a )
& ! [V7n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V7n ) ) @ ( ap @ V6f @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) )
& ! [V8n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ V1R @ ( inj__ty_2Enum_2Enum @ V8n ) ) @ ( ap @ V6f @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) @ ( ap @ V6f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDEPENDENT__CHOICE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ( ( ? [V2a: $i] :
( ( mem @ V2a @ A_27a )
& ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V2a ) ) )
& ! [V3n: tp__ty_2Enum_2Enum,V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V4x ) )
=> ? [V5y: $i] :
( ( mem @ V5y @ A_27a )
& ( p @ ( ap @ ( ap @ V0P @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V5y ) )
& ( p @ ( ap @ ( ap @ ( ap @ V1R @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V4x ) @ V5y ) ) ) ) ) )
=> ? [V6f: $i] :
( ( mem @ V6f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ! [V7n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V7n ) ) @ ( ap @ V6f @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) )
& ! [V8n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ V1R @ ( inj__ty_2Enum_2Enum @ V8n ) ) @ ( ap @ V6f @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) @ ( ap @ V6f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBIGUNION__MONO__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2g @ V3x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( arr @ A_27b @ bool ) ) @ V1f ) @ V0s ) ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( arr @ A_27b @ bool ) ) @ V2g ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBIGUNION__MONO,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2x ) @ V0s ) )
=> ? [V3y: $i] :
( ( mem @ V3y @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3y ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2x ) @ V3y ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__WLOG__LT,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ( ! [V1x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
<=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal,V5y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) ) )
=> ! [V6x: tp__ty_2Erealax_2Ereal,V7y: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V7y ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Edist,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__REFL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__SYM,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__ALT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__EQ__0,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V0x = V1y ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__POS__LT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( V0x != V1y )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__NZ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( V0x != V1y )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal,V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__LT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal,V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__HALF__L,axiom,
! [V0e: tp__ty_2Erealax_2Ereal,V1x1: tp__ty_2Erealax_2Ereal,V2x2: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x1 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x2 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x1 ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x2 ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__HALF__R,axiom,
! [V0e: tp__ty_2Erealax_2Ereal,V1x1: tp__ty_2Erealax_2Ereal,V2x2: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x1 ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x2 ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x1 ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x2 ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__ADD,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1x_27: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal,V3y_27: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y_27 ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x_27 ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y_27 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__MUL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__ADD__HALF,axiom,
! [V0e: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal,V2x_27: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal,V4y_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x_27 ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y_27 ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V2x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y_27 ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__LE__0,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V0x = V1y ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__POS__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__EQ,axiom,
! [V0w: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal,V3z: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0w ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0w ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__0,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__CHOOSE__DIST,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
=> ? [V2y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
= V1e ) ) ).
thf(ax_thm_2Ereal__topology_2Elinear,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
<=> ( ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) )
& ! [V3c: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__SCALING,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2Elinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ V1x ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__CMUL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Elinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ V0f @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__NEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Elinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__ADD,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1g ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Elinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V2x ) ) @ ( ap @ V1g @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__SUB,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1g ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Elinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V2x ) ) @ ( ap @ V1g @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1g ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Elinear @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__ID,axiom,
( p
@ ( ap @ c_2Ereal__topology_2Elinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] : V0x ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__ZERO,axiom,
( p
@ ( ap @ c_2Ereal__topology_2Elinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__NEGATION,axiom,
( p
@ ( ap @ c_2Ereal__topology_2Elinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V0x ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__SUM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V1s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Elinear @ ( ap @ V0f @ V2a ) ) ) ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Elinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] :
( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V4a: $i] : ( ap @ ( ap @ V0f @ V4a ) @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__MUL__COMPONENT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1v: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Elinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1v ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__0,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__CMUL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__NEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__ADD,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__SUB,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__SUM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1g ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__SUM__MUL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3v: $i] :
( ( mem @ V3v @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ V0f
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V4i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2c @ V4i ) ) @ ( ap @ V3v @ V4i ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V5i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2c @ V5i ) ) @ ( ap @ V0f @ ( ap @ V3v @ V5i ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__BOUNDED,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ? [V1B: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1B ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__BOUNDED__POS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ? [V1B: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1B ) ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1B ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESYMMETRIC__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V1s ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) ) )
=> ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ebilinear,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0f ) )
<=> ( ! [V1x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2Elinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2y ) ) ) )
& ! [V3y: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2Elinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ V0f @ V4x ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__SWAP,axiom,
! [V0op: $i] :
( ( mem @ V0op @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ( ( p
@ ( ap @ c_2Ereal__topology_2Ebilinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ ( ap @ V0op @ V2y ) @ V1x ) ) ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0op ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__LADD,axiom,
! [V0h: $i] :
( ( mem @ V0h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal,V3z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0h ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0h @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__RADD,axiom,
! [V0h: $i] :
( ( mem @ V0h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal,V3z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0h ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__LMUL,axiom,
! [V0h: $i] :
( ( mem @ V0h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0h ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0h @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__RMUL,axiom,
! [V0h: $i] :
( ( mem @ V0h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0h ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__LNEG,axiom,
! [V0h: $i] :
( ( mem @ V0h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0h ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0h @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__RNEG,axiom,
! [V0h: $i] :
( ( mem @ V0h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0h ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__LZERO,axiom,
! [V0h: $i] :
( ( mem @ V0h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0h ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0h @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__RZERO,axiom,
! [V0h: $i] :
( ( mem @ V0h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0h ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__LSUB,axiom,
! [V0h: $i] :
( ( mem @ V0h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal,V3z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0h ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0h @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__RSUB,axiom,
! [V0h: $i] :
( ( mem @ V0h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal,V3z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0h ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__SUM,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V4h: $i] :
( ( mem @ V4h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V4h ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V4h @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V1t ) @ V3g ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ty_2Erealax_2Ereal )
@ ( lam @ A_27a
@ ^ [V5i: $i] :
( lam @ A_27b
@ ^ [V6j: $i] : ( ap @ ( ap @ V4h @ ( ap @ V2f @ V5i ) ) @ ( ap @ V3g @ V6j ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__BOUNDED,axiom,
! [V0h: $i] :
( ( mem @ V0h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0h ) )
=> ? [V1B: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1B ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__BOUNDED__POS,axiom,
! [V0h: $i] :
( ( mem @ V0h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V0h ) )
=> ? [V1B: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1B ) ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ V0h @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1B ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__SUM__PARTIAL__SUC,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V2h ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5k: $i] : ( ap @ ( ap @ V2h @ ( ap @ V0f @ V5k ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V5k ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V1g @ V5k ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V2h @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ ( ap @ V2h @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6k: $i] : ( ap @ ( ap @ V2h @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V6k ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V0f @ V6k ) ) ) @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V6k ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__SUM__PARTIAL__PRE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V2h ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5k: $i] : ( ap @ ( ap @ V2h @ ( ap @ V0f @ V5k ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1g @ V5k ) ) @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V5k ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V2h @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) @ ( ap @ ( ap @ V2h @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6k: $i] : ( ap @ ( ap @ V2h @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V6k ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V0f @ V6k ) ) ) @ ( ap @ V1g @ V6k ) ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Esubspace,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V0s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0s ) )
& ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) @ V0s ) ) )
& ! [V3c: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ V0s ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Espan,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Espan @ V0s )
= ( ap @ ( ap @ ( c_2Etopology_2Ehull @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Esubspace ) @ V0s ) ) ) ).
thf(ax_thm_2Ereal__topology_2Edependent,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Edependent @ V0s ) )
<=> ? [V1a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ c_2Ereal__topology_2Espan @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Erealax_2Ereal ) @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eindependent,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V0s ) )
<=> ~ ( p @ ( ap @ c_2Ereal__topology_2Edependent @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__UNIV,axiom,
p @ ( ap @ c_2Ereal__topology_2Esubspace @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__IMP__NONEMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V0s ) )
=> ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__0,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__ADD,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ V2s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__MUL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1c: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ V2s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__SUB,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ V2s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__SUM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V3x ) ) @ V0s ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2t ) @ V1f ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__LINEAR__PREIMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1s ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Esubspace
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V2x ) ) @ V1s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__TRIVIAL,axiom,
p @ ( ap @ c_2Ereal__topology_2Esubspace @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__BIGINTER,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1s ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__0__SUBSPACE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1s ) ) )
=> ( ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) )
=> ( V2x = V3y ) )
<=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( V4x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__UNION__CHAIN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
| ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__SPAN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Espan @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) )
= ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__MONO,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__SPAN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__CLAUSES,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V1a: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal,V5s: $i] :
( ( mem @ V5s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V5s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V5s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V5s ) ) ) ) )
& ! [V6x: tp__ty_2Erealax_2Ereal,V7c: tp__ty_2Erealax_2Ereal,V8s: $i] :
( ( mem @ V8s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V8s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V7c ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V8s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__INDUCT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1h ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1h ) ) )
=> ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) )
=> ( p @ ( ap @ V1h @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__EMPTY,axiom,
( ( ap @ c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__NONZERO,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V0s ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDEPENDENT__MONO,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Edependent @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Edependent @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__SUBSPACE,axiom,
! [V0b: $i] :
( ( mem @ V0b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0b ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1s ) @ ( ap @ c_2Ereal__topology_2Espan @ V0b ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1s ) ) )
=> ( ( ap @ c_2Ereal__topology_2Espan @ V0b )
= V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__INDUCT__ALT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ V1h @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ! [V2c: tp__ty_2Erealax_2Ereal,V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
& ( p @ ( ap @ V1h @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) )
=> ( p @ ( ap @ V1h @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) ) )
=> ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) )
=> ( p @ ( ap @ V1h @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__SUPERSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__INC,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__UNION__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__UNIV,axiom,
( ( ap @ c_2Ereal__topology_2Espan @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__0,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__ADD,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__MUL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1c: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__MUL__EQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1c: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( V1c
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__NEG__EQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1s ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__SUB,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__SUM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V3x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2t ) @ V1f ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__ADD__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__EQ__SELF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Espan @ V0s )
= V0s )
<=> ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__SUBSET__SUBSPACE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESURJECTIVE__IMAGE__EQ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V2t ) )
=> ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( ( ap @ V0f @ V4x )
= V3y ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V5x ) ) @ V2t ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V1s ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s )
= V2t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__TRANSLATION__SELF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) ) )
=> ( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V2x ) ) )
@ V0s )
= V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__TRANSLATION__SELF__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V0s ) )
=> ( ( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V2x ) ) )
@ V0s )
= V0s )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__SUMS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1t ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Esubspace
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Espan @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ ( ap @ c_2Ereal__topology_2Espan @ V0s ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ ( ap @ c_2Ereal__topology_2Espan @ V1t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__EQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__ABS__EQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) )
<=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
| ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__EQ__LINV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= V0x )
<=> ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__EQ__RINV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
<=> ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EABS__TRIANGLE__EQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__EQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ecollinear,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ V0s ) )
<=> ? [V1u: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V0s ) ) )
=> ? [V4c: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V4c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1u ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__EMPTY,axiom,
p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__SING,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__SMALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V0s ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__3,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__LEMMA,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( V1y
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ? [V2c: tp__ty_2Erealax_2Ereal] :
( V1y
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__LEMMA__ALT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ? [V2c: tp__ty_2Erealax_2Ereal] :
( V1y
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__EQUAL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EMUL__CAUCHY__SCHWARZ__EQUAL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__3__EXPAND,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( ( V0a = V2c )
| ? [V3u: tp__ty_2Erealax_2Ereal] :
( V1b
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3u ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3u ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__TRIPLES,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( V1a != V2b )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ V0s ) ) ) )
<=> ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__4__3,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( V0a != V1b )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) )
<=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__3__TRANS,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( V1b != V2c ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ebetween,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBETWEEN__REFL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBETWEEN__REFL__EQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) )
<=> ( V1x = V0a ) ) ).
thf(conj_thm_2Ereal__topology_2EBETWEEN__SYM,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBETWEEN__ANTISYM,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) )
=> ( V0a = V1b ) ) ).
thf(conj_thm_2Ereal__topology_2EBETWEEN__TRANS,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBETWEEN__TRANS__2,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBETWEEN__ABS,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBETWEEN__IMP__COLLINEAR,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__BETWEEN__CASES,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__DIST__BETWEEN,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__1,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ V0s ) ) ) ).
thf(ax_thm_2Ereal__topology_2Emidpoint,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EMIDPOINT__REFL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= V0x ) ).
thf(conj_thm_2Ereal__topology_2EMIDPOINT__SYM,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__MIDPOINT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EMIDPOINT__EQ__ENDPOINT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
= V0a )
<=> ( V0a = V1b ) )
& ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
= V1b )
<=> ( V0a = V1b ) )
& ( ( V0a
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
<=> ( V0a = V1b ) )
& ( ( V1b
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
<=> ( V0a = V1b ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBETWEEN__MIDPOINT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EMIDPOINT__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__MIDPOINT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EMIDPOINT__COLLINEAR,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( V0a != V2c )
=> ( ( V1b
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) )
<=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__MONO,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__BREAKDOWN,axiom,
! [V0b: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0b ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1s ) ) ) )
=> ? [V3k: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3k ) ) @ ( inj__ty_2Erealax_2Ereal @ V0b ) ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Erealax_2Ereal ) @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V0b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__SPAN__INSERT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal__topology_2Espan @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ V2s ) ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ c_2Ereal__topology_2Espan @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__INSERT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V1s ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V1s ) ) @ ( ap @ c_2Ereal__topology_2Eindependent @ V1s ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ereal__topology_2Eindependent @ V1s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__EMPTY,axiom,
p @ ( ap @ c_2Ereal__topology_2Eindependent @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__SING,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
<=> ( V0x
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__STDBASIS,axiom,
( p
@ ( ap @ c_2Ereal__topology_2Eindependent
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0i: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V0i ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0i ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V0i ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPANNING__SUBSET__INDEPENDENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Espan @ V1t ) ) ) )
=> ( V0s = V1t ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__SPAN__DELETE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal__topology_2Espan @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Erealax_2Ereal ) @ V2s ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ c_2Ereal__topology_2Espan @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Erealax_2Ereal ) @ V2s ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__TRANS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Ereal__topology_2Espan @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2s ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEXCHANGE__LEMMA,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Espan @ V1t ) ) ) )
=> ? [V2t_27: $i] :
( ( mem @ V2t_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ ty_2Erealax_2Ereal ) @ V2t_27 ) @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V1t ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V2t_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2t_27 ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Espan @ V2t_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECARD__STDBASIS,axiom,
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__SPAN__BOUND,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Espan @ V1t ) ) ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__BOUND,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V0s ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V0s ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EMAXIMAL__INDEPENDENT__SUBSET__EXTEND,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1v ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V0s ) ) )
=> ? [V2b: $i] :
( ( mem @ V2b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V2b ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2b ) @ V1v ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V2b ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1v ) @ ( ap @ c_2Ereal__topology_2Espan @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EMAXIMAL__INDEPENDENT__SUBSET,axiom,
! [V0v: $i] :
( ( mem @ V0v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ? [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1b ) @ V0v ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0v ) @ ( ap @ c_2Ereal__topology_2Espan @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__BREAKDOWN__EQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1a: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V2s ) ) ) )
<=> ? [V3k: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3k ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__INDEPENDENT__EXTEND__LEMMA,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1b ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V1b ) )
=> ? [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1b ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1b ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) ) )
& ! [V5x: tp__ty_2Erealax_2Ereal,V6c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1b ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V6c ) ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V6c ) ) @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) ) )
& ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ V1b ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__INDEPENDENT__EXTEND,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V1b ) )
=> ? [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V2g ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1b ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__KERNEL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Esubspace
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V1x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__EQ__0__SPAN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1b ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
=> ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1b ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__EQ__0,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2s ) @ ( ap @ c_2Ereal__topology_2Espan @ V1b ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1b ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1g ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3s ) @ ( ap @ c_2Ereal__topology_2Espan @ V2b ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V2b ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) )
=> ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V3s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__EQ__STDBASIS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1g ) )
& ! [V2i: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2i ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2i ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2i ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V2i ) ) ) ) ) )
=> ( V0f = V1g ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__LEFT__INVERSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
=> ( V1x = V2y ) ) )
=> ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V3g ) )
& ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V3g ) @ V0f )
= ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : V4x ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Edim,axiom,
! [V0v: $i] :
( ( mem @ V0v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Ereal__topology_2Edim @ V0v ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ( arr @ ty_2Erealax_2Ereal @ bool )
@ ^ [V2b: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2b ) @ V0v ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ereal__topology_2Eindependent @ V2b ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0v ) @ ( ap @ c_2Ereal__topology_2Espan @ V2b ) ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ ty_2Erealax_2Ereal ) @ V2b ) @ V1n ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBASIS__EXISTS,axiom,
! [V0v: $i] :
( ( mem @ V0v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ? [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1b ) @ V0v ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0v ) @ ( ap @ c_2Ereal__topology_2Espan @ V1b ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ ty_2Erealax_2Ereal ) @ V1b ) @ ( ap @ c_2Ereal__topology_2Edim @ V0v ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECARD__GE__DIM__INDEPENDENT,axiom,
! [V0v: $i] :
( ( mem @ V0v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1b ) @ V0v ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V1b ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Ereal__topology_2Edim @ V0v ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V1b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0v ) @ ( ap @ c_2Ereal__topology_2Espan @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__EXPLICIT,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Espan @ V0p )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1y )
@ ( ap @ ( c_2Ebool_2E_3F @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ( arr @ ty_2Erealax_2Ereal @ bool )
@ ^ [V2s: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( lam @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ^ [V3u: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V2s ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2s ) @ V0p ) )
@ ( ap
@ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Erealax_2Ereal ) @ V2s )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4v: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V3u @ V4v ) ) @ V4v ) ) ) )
@ V1y ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDEPENDENT__EXPLICIT,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Edependent @ V0p ) )
<=> ? [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1s ) @ V0p ) )
& ? [V3v: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3v ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2u @ ( inj__ty_2Erealax_2Ereal @ V3v ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Erealax_2Ereal ) @ V1s )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4v: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2u @ V4v ) ) @ V4v ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__INJECTIVE__IMAGE__GEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( ap @ c_2Ereal__topology_2Espan @ V1s ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) )
=> ( V2x = V3y ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__INJECTIVE__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) )
=> ( V2x = V3y ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( ( ap @ c_2Ereal__topology_2Espan @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Espan @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__IMP__SURJECTIVE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
=> ( V1x = V2y ) ) )
=> ! [V3y: tp__ty_2Erealax_2Ereal] :
? [V4x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= V3y ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__INVERTIBLE__BOUNDED__BELOW__POS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1g ) )
& ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f )
= ( c_2Ecombin_2EI @ ty_2Erealax_2Ereal ) ) )
=> ? [V2B: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2B ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2B ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__INVERTIBLE__BOUNDED__BELOW,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1g ) )
& ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f )
= ( c_2Ecombin_2EI @ ty_2Erealax_2Ereal ) ) )
=> ? [V2B: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2B ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__BOUNDED__BELOW__POS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
=> ( V1x = V2y ) ) )
=> ? [V3B: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3B ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3B ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__CARD__LE__DIM,axiom,
! [V0v: $i] :
( ( mem @ V0v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1b ) @ V0v ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V1b ) ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1b ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V1b ) ) @ ( ap @ c_2Ereal__topology_2Edim @ V0v ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__CARD__GE__DIM,axiom,
! [V0v: $i] :
( ( mem @ V0v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0v ) @ ( ap @ c_2Ereal__topology_2Espan @ V1b ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1b ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Ereal__topology_2Edim @ V0v ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBASIS__CARD__EQ__DIM,axiom,
! [V0v: $i] :
( ( mem @ V0v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1b ) @ V0v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0v ) @ ( ap @ c_2Ereal__topology_2Espan @ V1b ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V1b ) ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1b ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V1b ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Ereal__topology_2Edim @ V0v ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBASIS__HAS__SIZE__DIM,axiom,
! [V0v: $i] :
( ( mem @ V0v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V1b ) )
& ( ( ap @ c_2Ereal__topology_2Espan @ V1b )
= V0v ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ ty_2Erealax_2Ereal ) @ V1b ) @ ( ap @ c_2Ereal__topology_2Edim @ V0v ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIM__UNIQUE,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1v: $i] :
( ( mem @ V1v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2b ) @ V1v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1v ) @ ( ap @ c_2Ereal__topology_2Espan @ V2b ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eindependent @ V2b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ ty_2Erealax_2Ereal ) @ V2b ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Ereal__topology_2Edim @ V1v ) )
= V0n ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIM__LE__CARD,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Ereal__topology_2Edim @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPAN__STDBASIS,axiom,
( ( ap @ c_2Ereal__topology_2Espan
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0i: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V0i ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0i ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V0i ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EHAS__SIZE__STDBASIS,axiom,
( p
@ ( ap
@ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0i: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V0i ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0i ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V0i ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
@ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIM__UNIV,axiom,
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Ereal__topology_2Edim @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIM__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Ereal__topology_2Edim @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Edim @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIM__SUBSET__UNIV,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Ereal__topology_2Edim @ V0s ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Esubtopology,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V1u )
= ( ap @ ( c_2Etopology_2Etopology @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2s: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2s ) @ V1u ) ) @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V2s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSET__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2t ) ) )
<=> ? [V3u: $i] :
( ( mem @ V3u @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3u ) @ V2t ) )
& ( V1s
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V3u ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTER__BIGUNION,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0s ) ) @ V1t )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2x ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2x ) @ V0s ) ) ) ) ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ V4t ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b ) @ V3s ) )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27b @ bool ) @ ( arr @ A_27b @ bool ) )
@ ( lam @ ( arr @ A_27b @ bool )
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ V4t ) @ V5x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V5x ) @ V3s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EISTOPLOGY__SUBTOPOLOGY,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ( p
@ ( ap @ ( c_2Etopology_2Eistopology @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2s: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2s ) @ V1u ) ) @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V2s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V1u ) ) @ V2s ) )
<=> ? [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V3t ) )
& ( V2s
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V3t ) @ V1u ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ETOPSPACE__SUBTOPOLOGY,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V1u ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) @ V1u ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V1u ) ) @ V2s ) )
<=> ? [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V3t ) )
& ( V2s
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V3t ) @ V1u ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__EMPTY,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V1s ) )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__EMPTY,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V1s ) )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__REFL,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V1u ) ) @ V1u ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1u ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__REFL,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V1u ) ) @ V1u ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1u ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBTOPOLOGY__SUPERSET,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) @ V1s ) )
=> ( ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V1s )
= V0top ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBTOPOLOGY__TOPSPACE,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) )
= V0top ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBTOPOLOGY__UNIV,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V0top ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__IMP__SUBSET,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V1s ) ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__IMP__SUBSET,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V1s ) ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__UNION,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V2t ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V3u ) ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V2t ) @ V3u ) ) ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__UNION,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V2t ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ V3u ) ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V2t ) @ V3u ) ) ) @ V1s ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eopen__def,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ! [V3x_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x_27 ) ) @ V0s ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eclosed__def,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eeuclidean,axiom,
( ( surj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean )
= ( surj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal @ ( ap @ ( c_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2EOpen ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__EMPTY,axiom,
p @ ( ap @ c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__UNIV,axiom,
p @ ( ap @ c_2Ereal__topology_2EOpen @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__BIGUNION,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1s ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__EXISTS__IN,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( p @ ( ap @ V0P @ V2a ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ V1Q @ V2a ) @ V3x ) ) ) ) ) ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V5a: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V5a ) ) @ ( ap @ ( ap @ V1Q @ V5a ) @ V4x ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__EXISTS,axiom,
! [A_27a: del,V0Q: $i] :
( ( mem @ V0Q @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ V0Q @ V1a ) @ V2x ) ) ) ) ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V4a: $i] : ( ap @ ( ap @ V0Q @ V4a ) @ V3x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ETOPSPACE__EUCLIDEAN,axiom,
( ( ap @ ( c_2Etopology_2Etopspace @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean )
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ETOPSPACE__EUCLIDEAN__SUBTOPOLOGY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ ( c_2Etopology_2Etopspace @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) )
= V0s ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__REFL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__REFL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__SUB__OPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ? [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2t ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__EMPTY,axiom,
p @ ( ap @ c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__UNIV,axiom,
p @ ( ap @ c_2Ereal__topology_2EClosed @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__BIGINTER,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBIGINTER__GSPEC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,A_27h: del,A_27i: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27b @ bool ) @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ bool ) @ ( ap @ V1f @ V2x ) ) @ ( ap @ V0P @ V2x ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V3a: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V3a )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V0P @ V4x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3a ) @ ( ap @ V1f @ V4x ) ) ) ) ) ) ) ) ) ) )
& ! [V5P: $i] :
( ( mem @ V5P @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27c @ ( arr @ A_27d @ ( arr @ A_27e @ bool ) ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27e )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27e @ bool ) @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ ( ty_2Epair_2Eprod @ ( arr @ A_27e @ bool ) @ bool ) )
@ ( lam @ A_27c
@ ^ [V7x: $i] :
( lam @ A_27d
@ ^ [V8y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27e @ bool ) @ bool ) @ ( ap @ ( ap @ V6f @ V7x ) @ V8y ) ) @ ( ap @ ( ap @ V5P @ V7x ) @ V8y ) ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27e @ A_27e )
@ ( lam @ A_27e
@ ^ [V9a: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27e @ bool ) @ V9a )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27c )
@ ( lam @ A_27c
@ ^ [V10x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27d )
@ ( lam @ A_27d
@ ^ [V11y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ V5P @ V10x ) @ V11y ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27e ) @ V9a ) @ ( ap @ ( ap @ V6f @ V10x ) @ V11y ) ) ) ) ) ) ) ) ) ) ) ) )
& ! [V12P: $i] :
( ( mem @ V12P @ ( arr @ A_27f @ ( arr @ A_27g @ ( arr @ A_27h @ bool ) ) ) )
=> ! [V13f: $i] :
( ( mem @ V13f @ ( arr @ A_27f @ ( arr @ A_27g @ ( arr @ A_27h @ ( arr @ A_27i @ bool ) ) ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27i )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27i @ bool ) @ ( ty_2Epair_2Eprod @ A_27f @ ( ty_2Epair_2Eprod @ A_27g @ A_27h ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27f @ ( ty_2Epair_2Eprod @ A_27g @ A_27h ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27i @ bool ) @ bool ) )
@ ( lam @ A_27f
@ ^ [V14x: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27g @ A_27h @ ( ty_2Epair_2Eprod @ ( arr @ A_27i @ bool ) @ bool ) )
@ ( lam @ A_27g
@ ^ [V15y: $i] :
( lam @ A_27h
@ ^ [V16z: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27i @ bool ) @ bool ) @ ( ap @ ( ap @ ( ap @ V13f @ V14x ) @ V15y ) @ V16z ) ) @ ( ap @ ( ap @ ( ap @ V12P @ V14x ) @ V15y ) @ V16z ) ) ) ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27i @ A_27i )
@ ( lam @ A_27i
@ ^ [V17a: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27i @ bool ) @ V17a )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27f )
@ ( lam @ A_27f
@ ^ [V18x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27g )
@ ( lam @ A_27g
@ ^ [V19y: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27h )
@ ( lam @ A_27h
@ ^ [V20z: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( ap @ V12P @ V18x ) @ V19y ) @ V20z ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27i ) @ V17a ) @ ( ap @ ( ap @ ( ap @ V13f @ V18x ) @ V19y ) @ V20z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__FORALL__IN,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( p @ ( ap @ V0P @ V2a ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ V1Q @ V2a ) @ V3x ) ) ) ) ) ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V5a: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V0P @ V5a ) ) @ ( ap @ ( ap @ V1Q @ V5a ) @ V4x ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__FORALL,axiom,
! [A_27a: del,V0Q: $i] :
( ( mem @ V0Q @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ V0Q @ V1a ) @ V2x ) ) ) ) ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V4a: $i] : ( ap @ ( ap @ V0Q @ V4a ) @ V3x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__DIFF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__DIFF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__BIGINTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) )
& ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__BIGUNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) )
& ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0s ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eball,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2y ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ecball,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2y ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Esphere,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2y ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__BALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__CBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__SPHERE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= V2e ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__BALL__0,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__CBALL__0,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__SPHERE__0,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= V1e ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__TRIVIAL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__TRIVIAL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECENTRE__IN__CBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__SUBSET__CBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPHERE__SUBSET__CBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSET__BALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1d: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSET__CBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1d: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__MAX__UNION,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal,V2s: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ c_2Ereal_2Emax @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) @ ( inj__ty_2Erealax_2Ereal @ V2s ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__MIN__INTER,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal,V2s: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ c_2Ereal_2Emin @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) @ ( inj__ty_2Erealax_2Ereal @ V2s ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__MAX__UNION,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal,V2s: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ c_2Ereal_2Emax @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) @ ( inj__ty_2Erealax_2Ereal @ V2s ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__MIN__INTER,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1d: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Emin @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__TRANSLATION,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal,V2r: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3y ) ) )
@ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__TRANSLATION,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal,V2r: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3y ) ) )
@ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPHERE__TRANSLATION,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal,V2r: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3y ) ) )
@ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2r: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V3y: tp__ty_2Erealax_2Ereal] :
? [V4x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= V3y )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2r: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V3y: tp__ty_2Erealax_2Ereal] :
? [V4x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= V3y )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPHERE__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2r: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V3y: tp__ty_2Erealax_2Ereal] :
? [V4x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= V3y )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__SCALING,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2r: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ V3x ) ) )
@ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__SCALING,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2r: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ V3x ) ) )
@ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__DIFF__BALL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) )
= ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__UNION__SPHERE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) )
= ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPHERE__UNION__BALL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) )
= ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__DIFF__SPHERE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) )
= ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__BALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECENTRE__IN__BALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__BALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__BALL__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
<=> ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__EQ__EMPTY,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__EMPTY,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__CBALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__CBALL__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
<=> ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPHERE__EQ__EMPTY,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPHERE__EMPTY,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENEGATIONS__BALL,axiom,
! [V0r: tp__ty_2Erealax_2Ereal] :
( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V1x ) ) )
@ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) ) )
= ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENEGATIONS__CBALL,axiom,
! [V0r: tp__ty_2Erealax_2Ereal] :
( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V1x ) ) )
@ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) ) )
= ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENEGATIONS__SPHERE,axiom,
! [V0r: tp__ty_2Erealax_2Ereal] :
( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V1x ) ) )
@ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) ) )
= ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1u ) ) @ V0s ) )
<=> ? [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) )
& ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1u ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__INTER__OPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN__INTER,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0u ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__OPEN__IN__TRANS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1u ) ) @ V0s ) )
<=> ? [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2t ) )
& ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1u ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__SUBSET__EQ,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ V1s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1s ) @ V0u ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__INTER__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED__INTER,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0u ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SUBSET__TRANS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBSET__TRANS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2Eopen__in,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ V1s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1s ) @ V0u ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
=> ? [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
& ! [V4x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) @ V0u ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) @ V1s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__CONTAINS__BALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ? [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) @ V1t ) ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__CONTAINS__CBALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ? [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) @ V1t ) ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__TRANS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__TRANS__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V2u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN__TRANS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__TRANS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__TRANS__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V2u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED__TRANS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__INTER__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1u ) @ V0s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2v ) @ V1u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2v ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2v ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) )
<=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) )
<=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eclosed__segment,axiom,
! [V0l: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0l ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1u: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V1u ) ) @ ( ap @ ( c_2Epair_2EFST @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( c_2Elist_2EHD @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0l ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ V1u ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( c_2Elist_2EHD @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0l ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1u ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V1u ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eopen__segment,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__SEGMENT__ALT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( V0a != V1b )
=> ( ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2u: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2u ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ V2u ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2u ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V2u ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2Esegment,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2u: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2u ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ V2u ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2u ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V2u ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
& ( ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEGMENT__REFL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
& ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__SEGMENT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ? [V3u: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3u ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3u ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( V2x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3u ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3u ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
<=> ( ( V0a != V1b )
& ? [V4u: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4u ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4u ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( V2x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4u ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V4u ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEGMENT__SYM,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
= ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EENDS__IN__SEGMENT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EENDS__NOT__IN__SEGMENT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEGMENT__CLOSED__OPEN,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEGMENT__OPEN__SUBSET__CLOSED,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EMIDPOINT__IN__SEGMENT,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) )
<=> ( V2a != V3b ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBETWEEN__IN__SEGMENT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ebetween @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__CONVEX__BOUND__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal,V3u: tp__ty_2Erealax_2Ereal,V4v: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3u ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4v ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V3u ) ) @ ( inj__ty_2Erealax_2Ereal @ V4v ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3u ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V4v ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__SEGMENT__COMPONENT,axiom,
! [A_27a: del,V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3i: $i] :
( ( mem @ V3i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Ereal_2Emin @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ c_2Ereal_2Emax @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEGMENT__TRANSLATION,axiom,
( ! [V0c: tp__ty_2Erealax_2Ereal,V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ V3x ) ) )
@ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ! [V4c: tp__ty_2Erealax_2Ereal,V5a: tp__ty_2Erealax_2Ereal,V6b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V4c ) ) @ ( inj__ty_2Erealax_2Ereal @ V5a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V4c ) ) @ ( inj__ty_2Erealax_2Ereal @ V6b ) ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V4c ) ) @ V7x ) ) )
@ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__SEGMENT__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__SEGMENT__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) )
=> ( V3x = V4y ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__OPEN__SEGMENT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( V2x != V0a )
& ( V2x != V1b ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__OPEN__SEGMENT__ALT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( V2x != V0a )
& ( V2x != V1b )
& ( V0a != V1b ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__DIST__IN__CLOSED__SEGMENT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__DIST__IN__OPEN__SEGMENT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ecollinear @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__IN__OPEN__CLOSED__SEGMENT,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) )
& ! [V3a: tp__ty_2Erealax_2Ereal,V4b: tp__ty_2Erealax_2Ereal,V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__IN__CLOSED__SEGMENT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__IN__OPEN__SEGMENT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Econnected,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ~ ? [V1e1: $i] :
( ( mem @ V1e1 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V2e2: $i] :
( ( mem @ V2e2 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1e1 ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2e2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 ) ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 ) ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V0s )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2e2 ) @ V0s )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ~ ? [V1e1: $i] :
( ( mem @ V1e1 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V2e2: $i] :
( ( mem @ V2e2 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1e1 ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2e2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 ) ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 ) ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V0s )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2e2 ) @ V0s )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__IN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ~ ? [V1e1: $i] :
( ( mem @ V1e1 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V2e2: $i] :
( ( mem @ V2e2 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1e1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V2e2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 ) ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1e1
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V2e2
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__IN__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ~ ? [V1e1: $i] :
( ( mem @ V1e1 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V2e2: $i] :
( ( mem @ V2e2 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1e1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V2e2 ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 )
= V0s )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1e1
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V2e2
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__IN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ~ ? [V1e1: $i] :
( ( mem @ V1e1 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V2e2: $i] :
( ( mem @ V2e2 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1e1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V2e2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 ) ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1e1
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V2e2
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__IN__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ~ ? [V1e1: $i] :
( ( mem @ V1e1 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V2e2: $i] :
( ( mem @ V2e2 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1e1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V2e2 ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 )
= V0s )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1e1
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V2e2
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEXISTS__DIFF,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ? [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V1s ) ) ) )
<=> ? [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ V0P @ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) ) )
=> ( ( V1t
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( V1t = V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__SET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ~ ? [V1e1: $i] :
( ( mem @ V1e1 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V2e2: $i] :
( ( mem @ V2e2 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1e1 ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2e2 ) )
& ( V1e1
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V2e2
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 )
= V0s )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__SET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ~ ? [V1e1: $i] :
( ( mem @ V1e1 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V2e2: $i] :
( ( mem @ V2e2 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1e1 ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2e2 ) )
& ( V1e1
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V2e2
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 )
= V0s )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1e1 ) @ V2e2 )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__IFF__CONNECTABLE__POINTS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ V0s ) ) )
=> ? [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3t ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ V3t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EMPTY,axiom,
p @ ( ap @ c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__REAL__LEMMA,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3e1: $i] :
( ( mem @ V3e1 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4e2: $i] :
( ( mem @ V4e2 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ V3e1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V4e2 ) )
& ! [V5e: tp__ty_2Erealax_2Ereal,V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) ) )
=> ? [V7d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7d ) ) )
& ! [V8y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V8y ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V8y ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) ) ) ) )
& ! [V9y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9y ) ) @ V3e1 ) )
=> ? [V10e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V10e ) ) )
& ! [V11y_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V11y_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V9y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V10e ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V11y_27 ) ) @ V3e1 ) ) ) ) )
& ! [V12y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12y ) ) @ V4e2 ) )
=> ? [V13e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V13e ) ) )
& ! [V14y_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V14y_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V12y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V13e ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V14y_27 ) ) @ V4e2 ) ) ) ) )
& ~ ? [V15x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V15x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V15x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V15x ) ) ) @ V3e1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V15x ) ) ) @ V4e2 ) ) ) )
=> ? [V16x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V16x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V16x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V16x ) ) ) @ V3e1 ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V16x ) ) ) @ V4e2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__SEGMENT,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__UNIV,axiom,
p @ ( ap @ c_2Ereal__topology_2Econnected @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( V0s
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__BIGUNION,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0P ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) ) ) )
& ( ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0P )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0P ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__DIFF__OPEN__FROM__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V2u ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V2u ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V2u ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__DISJOINT__BIGUNION__OPEN__UNIQUE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1f_27: $i] :
( ( mem @ V1f_27 @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epairwise @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( c_2Epred__set_2EDISJOINT @ ty_2Erealax_2Ereal ) ) @ V0f ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epairwise @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( c_2Epred__set_2EDISJOINT @ ty_2Erealax_2Ereal ) ) @ V1f_27 ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V0f ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V2s ) )
& ( V2s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3s ) @ V1f_27 ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V3s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V3s ) )
& ( V3s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
& ( ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0f )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V1f_27 ) ) )
=> ( V0f = V1f_27 ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__FROM__CLOSED__UNION__AND__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__FROM__OPEN__UNION__AND__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__INDUCTION,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V2s ) )
& ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ V3t ) ) )
=> ? [V5z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ V3t ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) ) ) )
& ! [V6a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ V2s ) )
=> ? [V7t: $i] :
( ( mem @ V7t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V7t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ V7t ) )
& ! [V8x: tp__ty_2Erealax_2Ereal,V9y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ V7t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9y ) ) @ V7t ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V9y ) ) )
& ( p @ ( ap @ V1Q @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) )
=> ( p @ ( ap @ V1Q @ ( inj__ty_2Erealax_2Ereal @ V9y ) ) ) ) ) ) )
=> ! [V10a: tp__ty_2Erealax_2Ereal,V11b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V10a ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V11b ) ) @ V2s ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V10a ) ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V11b ) ) )
& ( p @ ( ap @ V1Q @ ( inj__ty_2Erealax_2Ereal @ V10a ) ) ) )
=> ( p @ ( ap @ V1Q @ ( inj__ty_2Erealax_2Ereal @ V11b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EQUIVALENCE__RELATION__GEN,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V2s ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ V1R @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) )
=> ( p @ ( ap @ ( ap @ V1R @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
& ! [V5x: tp__ty_2Erealax_2Ereal,V6y: tp__ty_2Erealax_2Ereal,V7z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ V1R @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) )
& ( p @ ( ap @ ( ap @ V1R @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) @ ( inj__ty_2Erealax_2Ereal @ V7z ) ) ) )
=> ( p @ ( ap @ ( ap @ V1R @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V7z ) ) ) )
& ! [V8t: $i] :
( ( mem @ V8t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V9a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V8t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9a ) ) @ V8t ) ) )
=> ? [V10z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V10z ) ) @ V8t ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V10z ) ) ) ) ) )
& ! [V11a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V11a ) ) @ V2s ) )
=> ? [V12t: $i] :
( ( mem @ V12t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V12t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V11a ) ) @ V12t ) )
& ! [V13x: tp__ty_2Erealax_2Ereal,V14y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V13x ) ) @ V12t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V14y ) ) @ V12t ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V13x ) ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V14y ) ) ) )
=> ( p @ ( ap @ ( ap @ V1R @ ( inj__ty_2Erealax_2Ereal @ V13x ) ) @ ( inj__ty_2Erealax_2Ereal @ V14y ) ) ) ) ) ) )
=> ! [V15a: tp__ty_2Erealax_2Ereal,V16b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V15a ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V16b ) ) @ V2s ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V15a ) ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V16b ) ) ) )
=> ( p @ ( ap @ ( ap @ V1R @ ( inj__ty_2Erealax_2Ereal @ V15a ) ) @ ( inj__ty_2Erealax_2Ereal @ V16b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__INDUCTION__SIMPLE,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) )
& ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ V1s ) )
=> ? [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ V3t ) )
& ! [V4x: tp__ty_2Erealax_2Ereal,V5y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V3t ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
=> ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) ) ) ) )
=> ! [V6a: tp__ty_2Erealax_2Ereal,V7b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) @ V1s ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) ) )
=> ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EQUIVALENCE__RELATION,axiom,
! [V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal,V5y: tp__ty_2Erealax_2Ereal,V6z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) )
& ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) ) )
& ! [V7a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7a ) ) @ V1s ) )
=> ? [V8t: $i] :
( ( mem @ V8t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V8t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7a ) ) @ V8t ) )
& ! [V9x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) @ V8t ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V7a ) ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) ) ) ) ) )
=> ! [V10a: tp__ty_2Erealax_2Ereal,V11b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V10a ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V11b ) ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V10a ) ) @ ( inj__ty_2Erealax_2Ereal @ V11b ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Elimit__point__of,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) )
<=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2t ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) ) )
=> ? [V3y: tp__ty_2Erealax_2Ereal] :
( ( V3y != V0x )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__SUBSET,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1s ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__APPROACHABLE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3x_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x_27 ) ) @ V1s ) )
& ( V3x_27 != V0x )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EAPPROACHABLE__LT__LE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ? [V2d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) )
=> ( p @ ( ap @ V0P @ V3x ) ) ) ) )
<=> ? [V4d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
=> ( p @ ( ap @ V0P @ V5x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__APPROACHABLE__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3x_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x_27 ) ) @ V1s ) )
& ( V3x_27 != V0x )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__CHOOSE__SIZE,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) )
=> ? [V1x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= V0c ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__UNIV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__LIMPT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__EMPTY,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
~ ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENO__LIMIT__POINT__IMP__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ~ ? [V1x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__POSITIVE__ORTHANT,axiom,
( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V0x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__SET__AVOID,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1s ) )
=> ? [V2d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
& ( V3x != V0a ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMIT__POINT__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
=> ~ ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__SING,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
~ ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMIT__POINT__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
| ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__INSERT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__OF__LIMPTS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2y ) @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ V2y ) @ V1s ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__LIMPTS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ V1x ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDISCRETE__IMP__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) )
=> ( V3y = V2x ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__OF__UNIV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__OF__OPEN__IN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__OF__OPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SING,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Einterior,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ V0s )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x )
@ ( ap @ ( c_2Ebool_2E_3F @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ( arr @ ty_2Erealax_2Ereal @ bool )
@ ^ [V2t: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V1x ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2t ) @ V0s ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Einterior @ V0s )
= V0s )
<=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__OPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ V0s )
= V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__EMPTY,axiom,
( ( ap @ c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__UNIV,axiom,
( ( ap @ c_2Ereal__topology_2Einterior @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERIOR,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__INTERIOR,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) )
= ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) @ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSET__INTERIOR__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSET__INTERIOR,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__MAXIMAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__MAXIMAL__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1t ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__UNIQUE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) )
& ! [V2t_27: $i] :
( ( mem @ V2t_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2t_27 ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t_27 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2t_27 ) @ V1t ) ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ V0s )
= V1t ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__INTERIOR,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1s ) ) )
<=> ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__SUBSET__INTERIOR,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1t ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__FINITE__BIGINTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0s ) )
= ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ c_2Ereal__topology_2Einterior ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__BIGINTER__SUBSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0f ) ) ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ c_2Ereal__topology_2Einterior ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNION__INTERIOR__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__EQ__EMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Einterior @ V0s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) ) )
=> ( V1t
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__EQ__EMPTY__ALT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Einterior @ V0s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__LIMIT__POINT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CLOSED__UNION__EMPTY__INTERIOR,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( ( ap @ c_2Ereal__topology_2Einterior @ V1t )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
= ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__UNION__EQ__EMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
| ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( ( ap @ c_2Ereal__topology_2Einterior @ V0s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ c_2Ereal__topology_2Einterior @ V1t )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__UNIONS__OPEN__SUBSETS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ( arr @ ty_2Erealax_2Ereal @ bool )
@ ^ [V1t: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ V1t ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) ) ) ) ) )
= ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__ARCH__INV,axiom,
! [V0e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) )
<=> ? [V1n: tp__ty_2Enum_2Enum] :
( ( V1n != fo__c_2Enum_2E0 )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__POW__LBOUND,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__ARCH__POW,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ? [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__ARCH__POW2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
? [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__ARCH__POW__INV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ? [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFORALL__POS__MONO,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V1d: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) )
=> ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] :
( ( V3n != fo__c_2Enum_2E0 )
=> ( p @ ( ap @ V0P @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) )
=> ! [V4e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
=> ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFORALL__SUC,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ! [V1n: tp__ty_2Enum_2Enum] :
( ( V1n != fo__c_2Enum_2E0 )
=> ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0P @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELT__NZ,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
<=> ( V0n != fo__c_2Enum_2E0 ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__ARCH__RDIV__EQ__0,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) )
& ! [V2m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) ) )
=> ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eclosure,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ V1x ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__APPROACHABLE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__NONEMPTY__OPEN__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) )
<=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2t ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V2t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__INTERIOR,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__HULL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ V0s )
= ( ap @ ( ap @ ( c_2Etopology_2Ehull @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2EClosed ) @ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Eclosure @ V0s )
= V0s )
<=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ V0s )
= V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) )
= ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSET__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__INTER__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__BIGINTER__SUBSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0f ) ) ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ c_2Ereal__topology_2Eclosure ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__MINIMAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__MINIMAL__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ V1t ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__UNIQUE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) )
& ! [V2t_27: $i] :
( ( mem @ V2t_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V2t_27 ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2t_27 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V2t_27 ) ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ V0s )
= V1t ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__EMPTY,axiom,
( ( ap @ c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__UNIV,axiom,
( ( ap @ c_2Ereal__topology_2Eclosure @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__BIGUNION,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0f ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0f ) )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ( arr @ ty_2Erealax_2Ereal @ bool )
@ ^ [V1s: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__EQ__EMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Eclosure @ V0s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__SUBSET__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ V0s ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__INTER__CLOSURE__EQ__EMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__IN__INTER__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V2u ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) )
= ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__INTER__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) )
= ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__INTER__CLOSURE__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__INTER__SUPERSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
= ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__COMPLEMENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__COMPLEMENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__INTERMEDIATE__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__UNION__STRONG,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__DIFF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__OF__CLOSURE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__LIMPT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__INTER__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) )
<=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) )
= V1t ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CLOSURE__IDEMP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) )
= ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__INTERIOR__IDEMP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) )
= ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENOWHERE__DENSE__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENOWHERE__DENSE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2u ) )
& ( V2u
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2u ) @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2u ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CLOSURE__INTER__OPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__INTERIOR__UNION__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREGULAR__OPEN__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) )
= V0s )
& ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) )
= V1t ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREGULAR__CLOSED__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) )
= V0s )
& ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ V1t ) )
= V1t ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREGULAR__CLOSED__BIGUNION,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0f ) )
& ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V0f ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ V1t ) )
= V1t ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0f ) ) )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0f ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIFF__CLOSURE__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDENSE__OPEN__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V2u ) ) )
| ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V2u ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2u ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2u ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2u ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Efrontier,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Efrontier @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__CLOSURES,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Efrontier @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__STRADDLE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V1s ) ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ( ? [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) )
& ? [V4x: tp__ty_2Erealax_2Ereal] :
( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__SUBSET__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__EMPTY,axiom,
( ( ap @ c_2Ereal__topology_2Efrontier @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__UNIV,axiom,
( ( ap @ c_2Ereal__topology_2Efrontier @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__SUBSET__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) @ V0s ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__COMPLEMENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) )
= ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__DISJOINT__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__INTER__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__UNION__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__INTERIORS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Efrontier @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__FRONTIER__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__FRONTIER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__FRONTIER__EMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
| ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__FRONTIER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
| ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) )
=> ( ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) )
= ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__FRONTIER__FRONTIER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) )
= ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNION__FRONTIER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V1t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__INTER__FRONTIER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V1t ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CLOSED__EQ__EMPTY__AS__FRONTIER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( ( ap @ c_2Ereal__topology_2Einterior @ V0s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
<=> ? [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) )
& ( V0s
= ( ap @ c_2Ereal__topology_2Efrontier @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
=> ( ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__UNION__FRONTIER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__INTERIOR__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__CLOSURE__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESET__DIFF__FRONTIER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) )
= ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__INTER__SUBSET__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V1t ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eisnet,axiom,
! [A_27a: del,A_27b: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Ereal__topology_2Eisnet @ A_27a @ A_27b ) @ V0g ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27b )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0g @ V3z ) @ V1x ) )
=> ( p @ ( ap @ ( ap @ V0g @ V3z ) @ V2y ) ) ) )
| ! [V4z: $i] :
( ( mem @ V4z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0g @ V4z ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ V0g @ V4z ) @ V1x ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Enet__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Ereal__topology_2Enet @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( ty_2Ereal__topology_2Enet @ A_27a ) ) @ ( c_2Ereal__topology_2Eisnet @ A_27a @ A_27a ) ) @ V0rep ) ) ) ).
thf(conj_thm_2Ereal__topology_2Enet__tybij,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( ap @ ( c_2Ereal__topology_2Emk__net @ A_27a ) @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ! [V4z: $i] :
( ( mem @ V4z @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V1r @ V4z ) @ V2x ) )
=> ( p @ ( ap @ ( ap @ V1r @ V4z ) @ V3y ) ) ) )
| ! [V5z: $i] :
( ( mem @ V5z @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V1r @ V5z ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1r @ V5z ) @ V2x ) ) ) ) ) ) )
<=> ( ( ap @ ( c_2Ereal__topology_2Enetord @ A_27b ) @ ( ap @ ( c_2Ereal__topology_2Emk__net @ A_27b ) @ V1r ) )
= V1r ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENET,axiom,
! [A_27a: del,V0n: $i] :
( ( mem @ V0n @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0n ) @ V3z ) @ V1x ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0n ) @ V3z ) @ V2y ) ) ) )
| ! [V4z: $i] :
( ( mem @ V4z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0n ) @ V4z ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0n ) @ V4z ) @ V1x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOLDNET,axiom,
! [A_27a: del,V0n: $i] :
( ( mem @ V0n @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0n ) @ V1x ) @ V1x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0n ) @ V2y ) @ V2y ) ) )
=> ? [V3z: $i] :
( ( mem @ V3z @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0n ) @ V3z ) @ V3z ) )
& ! [V4w: $i] :
( ( mem @ V4w @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0n ) @ V4w ) @ V3z ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0n ) @ V4w ) @ V1x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0n ) @ V4w ) @ V2y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENET__DILEMMA,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2net: $i] :
( ( mem @ V2net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( ? [V3a: $i] :
( ( mem @ V3a @ A_27a )
& ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V2net ) @ V4x ) @ V3a ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V2net ) @ V5x ) @ V3a ) )
=> ( p @ ( ap @ V0P @ V5x ) ) ) ) )
& ? [V6b: $i] :
( ( mem @ V6b @ A_27a )
& ? [V7x: $i] :
( ( mem @ V7x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V2net ) @ V7x ) @ V6b ) ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V2net ) @ V8x ) @ V6b ) )
=> ( p @ ( ap @ V1Q @ V8x ) ) ) ) ) )
=> ? [V9c: $i] :
( ( mem @ V9c @ A_27a )
& ? [V10x: $i] :
( ( mem @ V10x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V2net ) @ V10x ) @ V9c ) ) )
& ! [V11x: $i] :
( ( mem @ V11x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V2net ) @ V11x ) @ V9c ) )
=> ( ( p @ ( ap @ V0P @ V11x ) )
& ( p @ ( ap @ V1Q @ V11x ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eat,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
= ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal
@ ( ap @ ( c_2Ereal__topology_2Emk__net @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2y ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eat__infinity,axiom,
( ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ c_2Ereal__topology_2Eat__infinity )
= ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal
@ ( ap @ ( c_2Ereal__topology_2Emk__net @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V1y: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ c_2Ereal_2Eabs @ V0x ) ) @ ( ap @ c_2Ereal_2Eabs @ V1y ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eat__posinfinity,axiom,
( ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ c_2Ereal__topology_2Eat__posinfinity )
= ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal
@ ( ap @ ( c_2Ereal__topology_2Emk__net @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V1y: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ V0x ) @ V1y ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eat__neginfinity,axiom,
( ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ c_2Ereal__topology_2Eat__neginfinity )
= ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal
@ ( ap @ ( c_2Ereal__topology_2Emk__net @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V1y: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V0x ) @ V1y ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Esequentially,axiom,
( ( surj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ c_2Ereal__topology_2Esequentially )
= ( surj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum
@ ( ap @ ( c_2Ereal__topology_2Emk__net @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V0m: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ V0m ) @ V1n ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ewithin,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ A_27a ) @ V0net ) @ V1s )
= ( ap @ ( c_2Ereal__topology_2Emk__net @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0net ) @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ein__direction,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1v: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal__topology_2Ein__direction @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1v ) ) )
= ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2b: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2b )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3c: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V3c ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ V2b ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ V3c ) @ ( inj__ty_2Erealax_2Ereal @ V1v ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EAT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EAT__INFINITY,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eat__infinity ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EAT__POSINFINITY,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eat__posinfinity ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EAT__NEGINFINITY,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eat__neginfinity ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEQUENTIALLY,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ ty_2Enum_2Enum ) @ c_2Ereal__topology_2Esequentially ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EWITHIN,axiom,
! [A_27a: del,V0n: $i] :
( ( mem @ V0n @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ A_27a ) @ V0n ) @ V1s ) ) @ V2x ) @ V3y ) )
<=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0n ) @ V2x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__DIRECTION,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1v: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Ein__direction @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1v ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) )
& ? [V4c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4c ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V4c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1v ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EWITHIN__UNIV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) )
= ( surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EWITHIN__WITHIN,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ A_27a ) @ V0net ) @ V1s ) ) @ V2t )
= ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ A_27a ) @ V0net ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Etrivial__limit,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
<=> ( ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ( V1a = V2b ) ) )
| ? [V3a: $i] :
( ( mem @ V3a @ A_27a )
& ? [V4b: $i] :
( ( mem @ V4b @ A_27a )
& ( V3a != V4b )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0net ) @ V5x ) @ V3a ) )
& ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0net ) @ V5x ) @ V4b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__WITHIN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ V0s ) ) )
<=> ~ ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__INFINITY,axiom,
~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eat__infinity ) ) ).
thf(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__POSINFINITY,axiom,
~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eat__posinfinity ) ) ).
thf(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__NEGINFINITY,axiom,
~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eat__neginfinity ) ) ).
thf(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__SEQUENTIALLY,axiom,
~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ ty_2Enum_2Enum ) @ c_2Ereal__topology_2Esequentially ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__CLOSED__TRIVIAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V1s ) ) )
=> ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENONTRIVIAL__LIMIT__WITHIN,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
=> ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ A_27a ) @ V0net ) @ V1s ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eeventually,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ! [V1net: $i] :
( ( mem @ V1net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ V0p ) @ V1net ) )
<=> ( ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V1net ) )
| ? [V2y: $i] :
( ( mem @ V2y @ A_27a )
& ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V1net ) @ V3x ) @ V2y ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V1net ) @ V4x ) @ V2y ) )
=> ( p @ ( ap @ V0p @ V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__HAPPENS,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ V1p ) @ V0net ) )
=> ( ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
| ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V1p @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__WITHIN__LE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2p: $i] :
( ( mem @ V2p @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Erealax_2Ereal ) @ V2p ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ V0s ) ) )
<=> ? [V3d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) )
=> ( p @ ( ap @ V2p @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__WITHIN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2p: $i] :
( ( mem @ V2p @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Erealax_2Ereal ) @ V2p ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ V0s ) ) )
<=> ? [V3d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) )
=> ( p @ ( ap @ V2p @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__AT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1p: $i] :
( ( mem @ V1p @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Erealax_2Ereal ) @ V1p ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) )
<=> ? [V2d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) )
=> ( p @ ( ap @ V1p @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__SEQUENTIALLY,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Enum_2Enum ) @ V0p ) @ c_2Ereal__topology_2Esequentially ) )
<=> ? [V1N: tp__ty_2Enum_2Enum] :
! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1N ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ V0p @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__INFINITY,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Erealax_2Ereal ) @ V0p ) @ c_2Ereal__topology_2Eat__infinity ) )
<=> ? [V1b: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
=> ( p @ ( ap @ V0p @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__POSINFINITY,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Erealax_2Ereal ) @ V0p ) @ c_2Ereal__topology_2Eat__posinfinity ) )
<=> ? [V1b: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
=> ( p @ ( ap @ V0p @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__NEGINFINITY,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Erealax_2Ereal ) @ V0p ) @ c_2Ereal__topology_2Eat__neginfinity ) )
<=> ? [V1b: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
=> ( p @ ( ap @ V0p @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__INFINITY__POS,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Erealax_2Ereal ) @ V0p ) @ c_2Ereal__topology_2Eat__infinity ) )
<=> ? [V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
=> ( p @ ( ap @ V0p @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EALWAYS__EVENTUALLY,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ! [V1net: $i] :
( ( mem @ V1net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ V0p @ V2x ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ V0p ) @ V1net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__AND,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27a @ bool ) )
=> ! [V2q: $i] :
( ( mem @ V2q @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1p @ V3x ) ) @ ( ap @ V2q @ V3x ) ) ) )
@ V0net ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ V1p ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ V2q ) @ V0net ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__MONO,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27a @ bool ) )
=> ! [V2q: $i] :
( ( mem @ V2q @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V1p @ V3x ) )
=> ( p @ ( ap @ V2q @ V3x ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ V1p ) @ V0net ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ V2q ) @ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__MP,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27a @ bool ) )
=> ! [V2q: $i] :
( ( mem @ V2q @ ( arr @ A_27a @ bool ) )
=> ( ( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1p @ V3x ) ) @ ( ap @ V2q @ V3x ) ) ) )
@ V0net ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ V1p ) @ V0net ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ V2q ) @ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__FALSE,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : c_2Ebool_2EF ) )
@ V0net ) )
<=> ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__TRUE,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : c_2Ebool_2ET ) )
@ V0net ) )
<=> $true ) ) ).
thf(conj_thm_2Ereal__topology_2ENOT__EVENTUALLY,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ~ ( p @ ( ap @ V1p @ V2x ) ) )
& ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ V1p ) @ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__FORALL,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s ) )
& ( V2s
!= ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V4a: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4a ) @ V2s ) ) @ ( ap @ ( ap @ V1p @ V4a ) @ V3x ) ) ) ) ) )
@ V0net ) )
<=> ! [V5a: $i] :
( ( mem @ V5a @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5a ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ ( ap @ V1p @ V5a ) ) @ V0net ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFORALL__EVENTUALLY,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s ) )
& ( V2s
!= ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
=> ( ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3a ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a ) @ ( ap @ V1p @ V3a ) ) @ V0net ) ) ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V5a: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5a ) @ V2s ) ) @ ( ap @ ( ap @ V1p @ V5a ) @ V4x ) ) ) ) ) )
@ V0net ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Etendsto,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2net: $i] :
( ( mem @ V2net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ V2net ) )
<=> ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) )
@ V2net ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Elim__def,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__topology_2Elim__def @ A_27a ) @ V0net ) @ V1f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2l: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ V2l ) @ V0net ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2net: $i] :
( ( mem @ V2net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ V2net ) )
<=> ( ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V2net ) )
| ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4y: $i] :
( ( mem @ V4y @ A_27a )
& ? [V5x: $i] :
( ( mem @ V5x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V2net ) @ V5x ) @ V4y ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V2net ) @ V6x ) @ V4y ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ V3s ) ) )
<=> ! [V4e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
=> ? [V5d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5d ) ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ V3s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ V3s ) ) )
<=> ! [V4e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
=> ? [V5d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5d ) ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ V3s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__AT__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) )
<=> ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__AT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) )
<=> ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__AT__INFINITY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Eat__infinity ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3b: tp__ty_2Erealax_2Ereal] :
! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__AT__INFINITY__POS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Eat__infinity ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__AT__POSINFINITY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Eat__posinfinity ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3b: tp__ty_2Erealax_2Ereal] :
! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__AT__NEGINFINITY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Eat__neginfinity ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3b: tp__ty_2Erealax_2Ereal] :
! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__SEQUENTIALLY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Esequentially ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3N: tp__ty_2Enum_2Enum] :
! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3N ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__EVENTUALLY,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) ) )
@ V0net ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__POSINFINITY__SEQUENTIALLY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Eat__posinfinity ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ V0f @ ( ap @ c_2Ereal_2Ereal__of__num @ V2n ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1l ) )
@ c_2Ereal__topology_2Esequentially ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__INFINITY__POSINFINITY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Eat__infinity ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Eat__posinfinity ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__EMPTY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__SUBSET,axiom,
! [V0t: $i] :
( ( mem @ V0t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3a: tp__ty_2Erealax_2Ereal,V4s: $i] :
( ( mem @ V4s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) @ V4s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0t ) @ V4s ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) @ V0t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__UNION,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2l: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V3s ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V4t ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__UNION__UNIV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2l: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V3s ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V4t ) ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t )
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__COMPOSE__WITHIN,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4y: tp__ty_2Erealax_2Ereal,V5z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V6w: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V6w ) ) @ V3s ) ) @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V6w ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) ) ) )
@ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) @ V3s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2g ) @ V1f ) ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ V0net ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__COMPOSE__AT,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3y: tp__ty_2Erealax_2Ereal,V4z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V5w: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V5w ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) ) ) )
@ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2g ) @ V1f ) ) @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) @ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__AT__WITHIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ V3s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__OPEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ V3s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V3s ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ V3s ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Efrom,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V0n ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1m: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ bool ) @ V1m ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1m ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFROM__0,axiom,
( ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__FROM,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFROM__INTER__NUMSEG__GEN,axiom,
! [V0k: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Enum_2Enum @ bool ) ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFROM__INTER__NUMSEG__MAX,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2EMAX @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFROM__INTER__NUMSEG,axiom,
! [V0k: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINFINITE__FROM,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EWLOG__LT,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ( ! [V1m: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
<=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) )
& ! [V4m: tp__ty_2Enum_2Enum,V5n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) )
=> ! [V6m: tp__ty_2Enum_2Enum,V7y: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ ( inj__ty_2Enum_2Enum @ V7y ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELT__EXISTS,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ? [V2d: tp__ty_2Enum_2Enum] :
( V1n
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2d ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LT__EQ,axiom,
! [V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ! [V1x: tp__ty_2Enum_2Enum,V2y: tp__ty_2Enum_2Enum,V3z: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) )
& ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V2y ) ) @ ( inj__ty_2Enum_2Enum @ V3z ) ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V3z ) ) ) )
=> ( ! [V4m: tp__ty_2Enum_2Enum,V5n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) )
<=> ! [V6n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LT,axiom,
! [V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ( ! [V1x: tp__ty_2Enum_2Enum,V2y: tp__ty_2Enum_2Enum,V3z: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) )
& ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V2y ) ) @ ( inj__ty_2Enum_2Enum @ V3z ) ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V3z ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
=> ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__SEQUENTIAL__INJ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) )
<=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Erealax_2Ereal ) @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
& ! [V4m: tp__ty_2Enum_2Enum,V5n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V4m ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) )
<=> ( V4m = V5n ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__SEQUENTIAL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) )
<=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Erealax_2Ereal ) @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINFINITE__SUPERSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__OPEN__BALL__CBALL,axiom,
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
<=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2t ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V2t ) ) ) ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V3s ) )
<=> ! [V5e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V3s ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) ) ) ) ) ) ) )
& ! [V6s: $i] :
( ( mem @ V6s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ V6s ) )
<=> ! [V8e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V8e ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V6s ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ ( inj__ty_2Erealax_2Ereal @ V8e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__OPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
<=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2t ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__BALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__CBALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINFINITE__OPEN__IN,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ V1s ) )
& ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0u ) ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1s ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Econdensation__point__of,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econdensation__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) )
<=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2t ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__OF__SUBSET,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econdensation__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1s ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econdensation__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__IMP__LIMPT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econdensation__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__BALL__CBALL,axiom,
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econdensation__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econdensation__point__of @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V3s ) )
<=> ! [V5e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V3s ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__BALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econdensation__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__CBALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econdensation__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__LINEAR,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1h ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ V1h @ ( ap @ V2f @ V4x ) ) ) )
@ ( ap @ V1h @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__CONST,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( inj__ty_2Erealax_2Ereal @ V1a ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1a ) )
@ V0net ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__CMUL,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( ap @ V1f @ V4x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) )
@ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__CMUL__EQ,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( V3c
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( ap @ V1f @ V4x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) )
@ V0net ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__NEG,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V1f @ V3x ) ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) )
@ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__NEG__EQ,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V1f @ V3x ) ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) )
@ V0net ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__ADD,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal,V4m: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ V0net ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V1f @ V5x ) ) @ ( ap @ V2g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__ABS,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V1f @ V3x ) ) ) )
@ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) )
@ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__SUB,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal,V4m: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ V0net ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1f @ V5x ) ) @ ( ap @ V2g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__MAX,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal,V4m: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ V0net ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ereal_2Emax @ ( ap @ V1f @ V5x ) ) @ ( ap @ V2g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Emax @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__MIN,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal,V4m: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ V0net ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ereal_2Emin @ ( ap @ V1f @ V5x ) ) @ ( ap @ V2g @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Emin @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__NULL,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__ABS,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0net ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V1f @ V2x ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__CMUL__EQ,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( V2c
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ V1f @ V3x ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0net ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__CMUL,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0net ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ V1f @ V3x ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__ADD,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0net ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2g @ V3x ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__SUB,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0net ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2g @ V3x ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__COMPARISON,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V1f @ V3x ) ) ) @ ( ap @ V2g @ V3x ) ) ) )
@ V0net ) )
& ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ V2g @ V4x ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0net ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__COMPONENT,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2i: $i] :
( ( mem @ V2i @ A_27b )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V4a: $i] : ( ap @ V1f @ V4a ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V3l ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__BOUND,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V1f @ V3n ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V2g @ V3n ) ) ) ) )
@ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0net ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__CMUL__BOUNDED,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3B: tp__ty_2Erealax_2Ereal] :
( ( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V4a: $i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V2g @ V4a ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V1f @ V4a ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3B ) ) ) ) )
@ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0net ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V5n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1f @ V5n ) ) @ ( ap @ V2g @ V5n ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__SUM,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27b ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3s ) )
& ! [V4i: $i] :
( ( mem @ V4i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4i ) @ V3s ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27b ) @ ( ap @ V1f @ V4i ) ) @ ( ap @ V2l @ V4i ) ) @ V0net ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27b )
@ ( lam @ A_27b
@ ^ [V5x: $i] :
( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V3s )
@ ( lam @ A_27a
@ ^ [V6i: $i] : ( ap @ ( ap @ V1f @ V6i ) @ V5x ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V3s ) @ V2l ) )
@ V0net ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__SUM,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s ) )
& ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3a ) @ V2s ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ V1f @ V4x ) @ V3a ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0net ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V2s ) @ ( ap @ V1f @ V5x ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__IN__CLOSED__SET,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2s ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V4x ) ) @ V2s ) ) )
@ V0net ) )
& ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__ABS__UBOUND,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V1f @ V4x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
@ V0net ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__ABS__LBOUND,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V1f @ V4x ) ) ) ) )
@ V0net ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__UNIQUE,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3l_27: tp__ty_2Erealax_2Ereal] :
( ( ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l_27 ) ) @ V0net ) ) )
=> ( V2l = V3l_27 ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ETENDSTO__LIM,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal] :
( ( ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__topology_2Elim__def @ A_27a ) @ V0net ) @ V1f ) )
= V2l ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__CONST__EQ,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2d: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( inj__ty_2Erealax_2Ereal @ V1c ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2d ) )
@ V0net ) )
<=> ( ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
| ( V1c = V2d ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORM__LIM__ADD,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ! [V4l: $i] :
( ( mem @ V4l @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V5m: $i] :
( ( mem @ V5m @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V6e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V7x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V8n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V8n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V2f @ V8n ) @ V7x ) ) @ ( ap @ V4l @ V8n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6e ) ) ) ) ) ) )
@ V0net ) ) )
& ! [V9e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V10x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V11n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V11n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V3g @ V11n ) @ V10x ) ) @ ( ap @ V5m @ V11n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) ) ) ) ) )
@ V0net ) ) ) )
=> ! [V12e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V12e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V13x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V14n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V14n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ V2f @ V14n ) @ V13x ) ) @ ( ap @ ( ap @ V3g @ V14n ) @ V13x ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V4l @ V14n ) ) @ ( ap @ V5m @ V14n ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V12e ) ) ) ) ) ) )
@ V0net ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORM__LIM__SUB,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ! [V4l: $i] :
( ( mem @ V4l @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V5m: $i] :
( ( mem @ V5m @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V6e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V7x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V8n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V8n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V2f @ V8n ) @ V7x ) ) @ ( ap @ V4l @ V8n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6e ) ) ) ) ) ) )
@ V0net ) ) )
& ! [V9e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V10x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V11n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V11n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V3g @ V11n ) @ V10x ) ) @ ( ap @ V5m @ V11n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) ) ) ) ) )
@ V0net ) ) ) )
=> ! [V12e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V12e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V13x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V14n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V14n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V2f @ V14n ) @ V13x ) ) @ ( ap @ ( ap @ V3g @ V14n ) @ V13x ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V4l @ V14n ) ) @ ( ap @ V5m @ V14n ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V12e ) ) ) ) ) ) )
@ V0net ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORM__LIM__BILINEAR,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27b @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27b @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ! [V5l: $i] :
( ( mem @ V5l @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V6m: $i] :
( ( mem @ V6m @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V7b1: tp__ty_2Erealax_2Ereal,V8b2: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V2h ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V9x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V10n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V10n ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V5l @ V10n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7b1 ) ) ) ) ) ) )
@ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V11x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V12n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V12n ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V6m @ V12n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V8b2 ) ) ) ) ) ) )
@ V0net ) )
& ! [V13e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V13e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V14x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V15n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V15n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V3f @ V15n ) @ V14x ) ) @ ( ap @ V5l @ V15n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V13e ) ) ) ) ) ) )
@ V0net ) ) )
& ! [V16e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V16e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V17x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V18n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V18n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V4g @ V18n ) @ V17x ) ) @ ( ap @ V6m @ V18n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V16e ) ) ) ) ) ) )
@ V0net ) ) ) )
=> ! [V19e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V19e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V20x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V21n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V21n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V2h @ ( ap @ ( ap @ V3f @ V21n ) @ V20x ) ) @ ( ap @ ( ap @ V4g @ V21n ) @ V20x ) ) ) @ ( ap @ ( ap @ V2h @ ( ap @ V5l @ V21n ) ) @ ( ap @ V6m @ V21n ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V19e ) ) ) ) ) ) )
@ V0net ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__BILINEAR,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V4l: tp__ty_2Erealax_2Ereal,V5m: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V4l ) ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V3g ) @ ( inj__ty_2Erealax_2Ereal @ V5m ) ) @ V0net ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V1h ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ ( ap @ V1h @ ( ap @ V2f @ V6x ) ) @ ( ap @ V3g @ V6x ) ) ) )
@ ( ap @ ( ap @ V1h @ ( inj__ty_2Erealax_2Ereal @ V4l ) ) @ ( inj__ty_2Erealax_2Ereal @ V5m ) ) )
@ V0net ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__ID,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : V2x ) )
@ ( inj__ty_2Erealax_2Ereal @ V0a ) )
@ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__AT__ID,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : V1x ) )
@ ( inj__ty_2Erealax_2Ereal @ V0a ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__AT__ZERO,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ V0f @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ V3x ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1l ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Enetlimit,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( ap @ ( c_2Ereal__topology_2Enetlimit @ A_27a ) @ V0net )
= ( ap @ ( c_2Emin_2E_40 @ A_27a )
@ ( lam @ A_27a
@ ^ [V1a: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2Enetord @ A_27a ) @ V0net ) @ V2x ) @ V1a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENETLIMIT__WITHIN,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ V1s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ereal__topology_2Enetlimit @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ V1s ) ) )
= V0a ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENETLIMIT__AT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ereal__topology_2Enetlimit @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) )
= V0a ) ).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1f @ V4x ) ) @ ( ap @ V2g @ V4x ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__EVENTUALLY,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V4x ) ) @ ( ap @ V2g @ V4x ) ) ) )
@ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN,axiom,
! [V0l: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3x: tp__ty_2Erealax_2Ereal,V4s: $i] :
( ( mem @ V4s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V5d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5d ) ) )
& ! [V6x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x_27 ) ) @ V4s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5d ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V6x_27 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V6x_27 ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V4s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V4s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__AT,axiom,
! [V0l: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3x: tp__ty_2Erealax_2Ereal,V4d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ! [V5x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__EQ,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1f @ V4x ) ) @ ( ap @ V2g @ V4x ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0net ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__SET,axiom,
! [V0l: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V3s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V4t ) ) ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ V3s ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ V4t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__SET__IMP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V4t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V3s ) ) ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ V3s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ V4t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__AWAY__WITHIN,axiom,
! [V0l: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal,V4b: tp__ty_2Erealax_2Ereal,V5s: $i] :
( ( mem @ V5s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( V3a != V4b )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ V5s ) )
& ( V6x != V3a )
& ( V6x != V4b ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) @ V5s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) @ V5s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__AWAY__AT,axiom,
! [V0l: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal,V4b: tp__ty_2Erealax_2Ereal] :
( ( ( V3a != V4b )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( V5x != V3a )
& ( V5x != V4b ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__OPEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal,V4l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ V2s ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V2s ) )
& ( V5x != V3a ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V4l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V4l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__OPEN__IN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4a: tp__ty_2Erealax_2Ereal,V5l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ V2s ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ V2s ) )
& ( V6x != V4a ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V5l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) ) @ V3t ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V5l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) ) @ V3t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__CASES__FINITE__SEQUENTIALLY,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ bool ) @ V4n ) @ ( ap @ V0P @ V4n ) ) ) ) ) )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ V0P @ V5n ) ) @ ( ap @ V1f @ V5n ) ) @ ( ap @ V2g @ V5n ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V3l ) )
@ c_2Ereal__topology_2Esequentially ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__CASES__COFINITE__SEQUENTIALLY,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ bool ) @ V4n ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ V0P @ V4n ) ) ) ) ) ) )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ V0P @ V5n ) ) @ ( ap @ V1f @ V5n ) ) @ ( ap @ V2g @ V5n ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V3l ) )
@ c_2Ereal__topology_2Esequentially ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__CASES__SEQUENTIALLY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Enum_2Enum] :
( ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ V4n ) ) @ ( ap @ V0f @ V4n ) ) @ ( ap @ V1g @ V4n ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2l ) )
@ c_2Ereal__topology_2Esequentially ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ c_2Ereal__topology_2Esequentially ) ) )
& ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ V5n ) ) @ ( ap @ V0f @ V5n ) ) @ ( ap @ V1g @ V5n ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2l ) )
@ c_2Ereal__topology_2Esequentially ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ c_2Ereal__topology_2Esequentially ) ) )
& ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V6n ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V0f @ V6n ) ) @ ( ap @ V1g @ V6n ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2l ) )
@ c_2Ereal__topology_2Esequentially ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ c_2Ereal__topology_2Esequentially ) ) )
& ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V7n ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V0f @ V7n ) ) @ ( ap @ V1g @ V7n ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2l ) )
@ c_2Ereal__topology_2Esequentially ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__CONG__WITHIN,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal,V4s: $i] :
( ( mem @ V4s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( V5x != V0a )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ V1f @ V6x ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V3l ) )
@ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ V4s ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ V4s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__CONG__AT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( V4x != V0a )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ V1f @ V5x ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V3l ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__SEQUENTIAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V2x @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__CONTAINS__SEQUENTIAL__LIMIT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1x @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ c_2Ereal__topology_2Esequentially ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__SEQUENTIAL__LIMITS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal] :
( ( ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1x @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ c_2Ereal__topology_2Esequentially ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__APPROACHABLE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) )
=> ( ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__CLOSURE__DELETE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Erealax_2Ereal ) @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDENSE__IMP__PERFECT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Eclosure @ V0s )
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDENSE__LIMIT__POINTS,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ V2x ) @ V0s ) ) ) )
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) )
<=> ( ( ap @ c_2Ereal__topology_2Eclosure @ V0s )
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEQ__OFFSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Esequentially ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V3i ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1l ) )
@ c_2Ereal__topology_2Esequentially ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEQ__OFFSET__NEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Esequentially ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V3i ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1l ) )
@ c_2Ereal__topology_2Esequentially ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEQ__OFFSET__REV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2k: tp__ty_2Enum_2Enum] :
( ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V3i ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1l ) )
@ c_2Ereal__topology_2Esequentially ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEQ__HARMONIC__OFFSET,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ V1n ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ c_2Ereal__topology_2Esequentially ) ) ).
thf(conj_thm_2Ereal__topology_2ESEQ__HARMONIC,axiom,
( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V0n: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ V0n ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ c_2Ereal__topology_2Esequentially ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__CBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__INTERIOR__CBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1s ) ) )
<=> ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__BALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__BALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) )
= ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__BALL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) )
= ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) )
= ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__BALL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
=> ( ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) )
= ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__CBALL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) )
= ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__EQ__EMPTY,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__EMPTY,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__EQ__SING,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
<=> ( V1e
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__SING,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( V1e
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPHERE__SING,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( V1e
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPHERE__EQ__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
<=> ( ( V2x = V0a )
& ( V1r
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__WITHIN__INTERIOR,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1s ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Erealax_2Ereal ) @ V0p ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V1s ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Erealax_2Ereal ) @ V0p ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__INTERIOR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V2s ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V2s ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENETLIMIT__WITHIN__INTERIOR,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Ereal__topology_2Enetlimit @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V0s ) ) )
= V1x ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__IMP__PERFECT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ~ ? [V2a: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__IMP__PERFECT__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ~ ? [V2a: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ebounded__def,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
<=> ? [V1a: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__EMPTY,axiom,
p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__INTERIOR,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CLOSURE__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__BALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__IMP__BOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
<=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__BIGUNION,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0f ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__POS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
<=> ? [V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__POS__LT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
<=> ? [V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
| ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__DIFF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__INSERT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__BIGINTER,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ? [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENOT__BOUNDED__UNIV,axiom,
~ ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOBOUNDED__IMP__UNBOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) ) )
=> ~ ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SCALING,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ V2x ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__NEGATIONS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V1x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__TRANSLATION,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__TRANSLATION__EQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) )
@ V1s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__DIFFS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUMS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUMS__IMAGE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ V0f @ V3x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2t ) ) ) ) ) )
& ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ V1g @ V4x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2t ) ) ) ) ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUMS__IMAGES,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s ) )
& ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3a ) @ V2s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ V0f @ V4x ) @ V3a ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1t ) ) ) ) ) ) ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V2s ) @ ( ap @ V0f @ V5x ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V1t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__BALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ? [V2r: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CBALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ? [V2r: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNBOUNDED__INTER__COBOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ~ ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V1t ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOBOUNDED__INTER__UNBOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) ) )
& ~ ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__BOUNDED__EQ__TRIVIAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V0s ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
<=> ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__COMPONENTWISE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
<=> ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : V1x ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__HAS__SUP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal_2Esup @ V0s ) ) ) )
& ! [V2b: tp__ty_2Erealax_2Ereal] :
( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Esup @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUP__INSERT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Esup @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Emax @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Esup @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__HAS__INF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
& ! [V2b: tp__ty_2Erealax_2Ereal] :
( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINF__INSERT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eiterate_2Einf @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Emin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Eiterate_2Einf @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSET__BALLS,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1a_27: tp__ty_2Erealax_2Ereal,V2r: tp__ty_2Erealax_2Ereal,V3r_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3r_27 ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a_27 ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3r_27 ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
& ! [V4a: tp__ty_2Erealax_2Ereal,V5a_27: tp__ty_2Erealax_2Ereal,V6r: tp__ty_2Erealax_2Ereal,V7r_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5a_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V7r_27 ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5a_27 ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6r ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7r_27 ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
& ! [V8a: tp__ty_2Erealax_2Ereal,V9a_27: tp__ty_2Erealax_2Ereal,V10r: tp__ty_2Erealax_2Ereal,V11r_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V10r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9a_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V11r_27 ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V9a_27 ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V10r ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V11r_27 ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V10r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
& ! [V12a: tp__ty_2Erealax_2Ereal,V13a_27: tp__ty_2Erealax_2Ereal,V14r: tp__ty_2Erealax_2Ereal,V15r_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12a ) ) @ ( inj__ty_2Erealax_2Ereal @ V14r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V13a_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V15r_27 ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12a ) ) @ ( inj__ty_2Erealax_2Ereal @ V13a_27 ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V14r ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V15r_27 ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V14r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTER__BALLS__EQ__EMPTY,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2r: tp__ty_2Erealax_2Ereal,V3s: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3s ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3s ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) @ ( inj__ty_2Erealax_2Ereal @ V3s ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) )
& ! [V4a: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal,V6r: tp__ty_2Erealax_2Ereal,V7s: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) @ ( inj__ty_2Erealax_2Ereal @ V7s ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V7s ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V6r ) ) @ ( inj__ty_2Erealax_2Ereal @ V7s ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) ) ) ) )
& ! [V8a: tp__ty_2Erealax_2Ereal,V9b: tp__ty_2Erealax_2Ereal,V10r: tp__ty_2Erealax_2Ereal,V11s: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V10r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) @ ( inj__ty_2Erealax_2Ereal @ V11s ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V10r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V11s ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V10r ) ) @ ( inj__ty_2Erealax_2Ereal @ V11s ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) ) ) ) ) )
& ! [V12a: tp__ty_2Erealax_2Ereal,V13b: tp__ty_2Erealax_2Ereal,V14r: tp__ty_2Erealax_2Ereal,V15s: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12a ) ) @ ( inj__ty_2Erealax_2Ereal @ V14r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V13b ) ) @ ( inj__ty_2Erealax_2Ereal @ V15s ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V14r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V15s ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V14r ) ) @ ( inj__ty_2Erealax_2Ereal @ V15s ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12a ) ) @ ( inj__ty_2Erealax_2Ereal @ V13b ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ecompact,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
<=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0s ) )
=> ? [V3l: tp__ty_2Erealax_2Ereal,V4r: $i] :
( ( mem @ V4r @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0s ) )
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V4r @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) @ ( ap @ V4r @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ V1f ) @ V4r ) ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EMONOTONE__BIGGER,axiom,
! [V0r: $i] :
( ( mem @ V0r @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V0r @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ V0r @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
=> ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ V0r @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__SUBSEQUENCE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal] :
( ( ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V1r @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V1r @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ c_2Ereal__topology_2Esequentially ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ V0s ) @ V1r ) ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EMONOTONE__SUBSEQUENCE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ? [V1r: $i] :
( ( mem @ V1r @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V1r @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ V1r @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
& ( ! [V4m: tp__ty_2Enum_2Enum,V5n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0s @ ( ap @ V1r @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) ) @ ( ap @ V0s @ ( ap @ V1r @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) ) )
| ! [V6m: tp__ty_2Enum_2Enum,V7n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ ( inj__ty_2Enum_2Enum @ V7n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0s @ ( ap @ V1r @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) ) @ ( ap @ V0s @ ( ap @ V1r @ ( inj__ty_2Enum_2Enum @ V6m ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONVERGENT__BOUNDED__INCREASING,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ( ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
=> ? [V5l: tp__ty_2Erealax_2Ereal] :
! [V6e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6e ) ) )
=> ? [V7N: tp__ty_2Enum_2Enum] :
! [V8n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V7N ) ) @ ( inj__ty_2Enum_2Enum @ V8n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONVERGENT__BOUNDED__MONOTONE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
& ( ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
| ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) ) ) ) )
=> ? [V7l: tp__ty_2Erealax_2Ereal] :
! [V8e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V8e ) ) )
=> ? [V9N: tp__ty_2Enum_2Enum] :
! [V10n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V9N ) ) @ ( inj__ty_2Enum_2Enum @ V10n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V10n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V8e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__REAL__LEMMA,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
=> ? [V3l: tp__ty_2Erealax_2Ereal,V4r: $i] :
( ( mem @ V4r @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V4r @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) @ ( ap @ V4r @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
& ! [V7e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7e ) ) )
=> ? [V8N: tp__ty_2Enum_2Enum] :
! [V9n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V8N ) ) @ ( inj__ty_2Enum_2Enum @ V9n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0s @ ( ap @ V4r @ ( inj__ty_2Enum_2Enum @ V9n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__LEMMA,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0x @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V1s ) ) )
=> ? [V3l: tp__ty_2Erealax_2Ereal,V4r: $i] :
( ( mem @ V4r @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V4r @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) @ ( ap @ V4r @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
& ! [V7e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7e ) ) )
=> ? [V8N: tp__ty_2Enum_2Enum] :
! [V9n: tp__ty_2Enum_2Enum,V10i: $i] :
( ( mem @ V10i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V8N ) ) @ ( inj__ty_2Enum_2Enum @ V9n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0x @ ( ap @ V4r @ ( inj__ty_2Enum_2Enum @ V9n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__IMP__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ecauchy,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ V0s ) )
<=> ! [V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
=> ? [V2N: tp__ty_2Enum_2Enum] :
! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V2N ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( inj__ty_2Enum_2Enum @ V2N ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ecomplete,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecomplete @ V0s ) )
<=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ V1f ) ) )
=> ? [V3l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECAUCHY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ V0s ) )
<=> ! [V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
=> ? [V2N: tp__ty_2Enum_2Enum] :
! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V2N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2N ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONVERGENT__IMP__CAUCHY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Esequentially ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EGREATER__EQ__REFL,axiom,
! [V0m: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUPPER__BOUND__FINITE__SET__REAL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ? [V2a: tp__ty_2Erealax_2Ereal] :
! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECAUCHY__IMP__BOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1y )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V1y ) @ ( ap @ V0s @ V2n ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__IMP__COMPLETE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecomplete @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPLETE__UNIV,axiom,
p @ ( ap @ c_2Ereal__topology_2Ecomplete @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPLETE__EQ__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecomplete @ V0s ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONVERGENT__EQ__CAUCHY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ? [V1l: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Esequentially ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONVERGENT__IMP__BOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Esequentially ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__IMP__TOTALLY__BOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ! [V1e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
=> ? [V2k: $i] :
( ( mem @ V2k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V2k ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2k ) @ V0s ) )
& ( p
@ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s )
@ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V3x ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) )
@ V2k ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHEINE__BOREL__LEMMA,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V1t ) ) )
& ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2b ) @ V1t ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2b ) ) ) ) )
=> ? [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
=> ? [V5b: $i] :
( ( mem @ V5b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V5b ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) @ V5b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__IMP__HEINE__BOREL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2t ) @ V1f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V1f ) ) ) )
=> ? [V3f_27: $i] :
( ( mem @ V3f_27 @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) @ V1f ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V3f_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHEINE__BOREL__IMP__BOLZANO__WEIERSTRASS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2t ) @ V1f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V1f ) ) ) )
=> ? [V3f_27: $i] :
( ( mem @ V3f_27 @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) @ V1f ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V3f_27 ) ) ) ) ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V4t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4t ) @ V0s ) ) )
=> ? [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V4t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__IMP__BOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) ) )
=> ? [V2x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1t ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINF__FINITE__LEMMA,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ V0s ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINF__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ V0s ) )
& ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LE__INF__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__INF__LE__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) )
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LT__INF__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__INF__LT__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) )
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEQUENCE__INFINITE__LEMMA,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( ! [V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
!= V1l )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Esequentially ) ) )
=> ~ ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3y )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V3y ) @ ( ap @ V0f @ V4n ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELE__1,axiom,
( ! [V0n: tp__ty_2Enum_2Enum] :
( ( V0n != fo__c_2Enum_2E0 )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( V1n != fo__c_2Enum_2E0 )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( V2n != fo__c_2Enum_2E0 ) )
& ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
& ! [V5n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
=> ( V5n != fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__OF__SEQUENCE__SUBSEQUENCE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
=> ? [V2r: $i] :
( ( mem @ V2r @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V2r @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V2r @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ V0f ) @ V2r ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEQUENCE__UNIQUE__LIMPT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2l_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Esequentially ) )
& ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2l_27 ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3y )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V3y ) @ ( ap @ V0f @ V4n ) ) ) ) ) ) ) ) ) )
=> ( V2l_27 = V1l ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__IMP__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) ) )
=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1t ) ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__EQ__HEINE__BOREL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
<=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2t ) @ V1f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V1f ) ) ) )
=> ? [V3f_27: $i] :
( ( mem @ V3f_27 @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) @ V1f ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V3f_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__EQ__BOLZANO__WEIERSTRASS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
<=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) ) )
=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__EQ__BOUNDED__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
<=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__IMP__BOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__IMP__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__SEQUENCE__WITH__LIMIT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ c_2Ereal__topology_2Esequentially ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__COMPACT__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) )
<=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__EQ__HEINE__BOREL__SUBTOPOLOGY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
<=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2t ) @ V1f ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V2t ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V1f ) ) ) )
=> ? [V3f_27: $i] :
( ( mem @ V3f_27 @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) @ V1f ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V3f_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__CONTRAPOS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ~ ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1t ) ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDISCRETE__BOUNDED__IMP__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) )
=> ( V3y = V2x ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) ) )
=> ? [V1x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__EQ__BOLZANO__WEIERSTRASS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
<=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) )
& ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1t ) ) )
=> ? [V2x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__EMPTY,axiom,
p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__INTER__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTER__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__BIGINTER,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) ) ) )
& ( V0f
!= ( c_2Epred__set_2EEMPTY @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__IMP__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__IMP__CLOSED__IN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__IMP__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__INSERT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__SING,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0u ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__SING,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__INSERT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__FRONTIER__BOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__FRONTIER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__FRONTIER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__SUBSET__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__DELETE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Erealax_2Ereal ) @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__DELETE,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Erealax_2Ereal ) @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__BIGINTER__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
<=> ! [V1e: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__BIGUNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) )
& ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__DIFF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__SPHERE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SPHERE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__SPHERE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__IMP__FIP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2t ) @ V1f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2t ) ) ) )
& ! [V3f_27: $i] :
( ( mem @ V3f_27 @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) @ V1f ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V3f_27 ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V1f ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IMP__FIP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2t ) @ V1f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2t ) ) ) )
& ? [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3t ) @ V1f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V3t ) ) )
& ! [V4f_27: $i] :
( ( mem @ V4f_27 @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V4f_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V4f_27 ) @ V1f ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V4f_27 ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V1f ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IMP__FIP__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2t ) @ V1f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V2t ) ) ) )
& ! [V3f_27: $i] :
( ( mem @ V3f_27 @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) @ V1f ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V3f_27 ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V1f ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__FIP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V0f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) ) )
& ? [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2t ) @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V2t ) ) )
& ! [V3f_27: $i] :
( ( mem @ V3f_27 @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3f_27 ) @ V0f ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V3f_27 )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0f )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__FIP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V0f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) ) ) )
& ! [V2f_27: $i] :
( ( mem @ V2f_27 @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2f_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2f_27 ) @ V0f ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V2f_27 )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0f )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__NEST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ( ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2n ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ? [V5a: tp__ty_2Erealax_2Ereal] :
! [V6n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5a ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDECREASING__CLOSED__NEST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ( ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2n ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
& ! [V5e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) )
=> ? [V6n: tp__ty_2Enum_2Enum] :
! [V7x: tp__ty_2Erealax_2Ereal,V8y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8y ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ ( inj__ty_2Erealax_2Ereal @ V8y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) ) ) ) )
=> ? [V9a: tp__ty_2Erealax_2Ereal] :
! [V10n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9a ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V10n ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDECREASING__CLOSED__NEST__SING,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ( ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2n ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
& ! [V5e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) )
=> ? [V6n: tp__ty_2Enum_2Enum] :
! [V7x: tp__ty_2Erealax_2Ereal,V8y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8y ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ ( inj__ty_2Erealax_2Ereal @ V8y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) ) ) ) )
=> ? [V9a: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ( arr @ ty_2Erealax_2Ereal @ bool )
@ ^ [V10t: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ V10t )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V11n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V10t ) @ ( ap @ V0s @ V11n ) ) ) ) ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__CHAIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V0f ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2s ) )
& ( V2s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3s ) @ V0f ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V4t ) @ V0f ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) )
| ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4t ) @ V3s ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1b ) @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1b ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0f )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CHAIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0f ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V0f ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3t ) @ V0f ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2s ) @ V3t ) )
| ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3t ) @ V2s ) ) ) ) ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0f )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__NEST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ( ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
& ( ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V1n ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( ap @ V0s @ V4n ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V4n ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONVERGENT__EQ__CAUCHY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ( ? [V2l: $i] :
( ( mem @ V2l @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
& ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4N: tp__ty_2Enum_2Enum] :
! [V5n: tp__ty_2Enum_2Enum,V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4N ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
& ( p @ ( ap @ V0P @ V6x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V6x ) ) @ ( ap @ V2l @ V6x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) )
<=> ! [V7e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7e ) ) )
=> ? [V8N: tp__ty_2Enum_2Enum] :
! [V9m: tp__ty_2Enum_2Enum,V10n: tp__ty_2Enum_2Enum,V11x: $i] :
( ( mem @ V11x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V8N ) ) @ ( inj__ty_2Enum_2Enum @ V9m ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V8N ) ) @ ( inj__ty_2Enum_2Enum @ V10n ) ) )
& ( p @ ( ap @ V0P @ V11x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V9m ) ) @ V11x ) ) @ ( ap @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V10n ) ) @ V11x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONVERGENT__EQ__CAUCHY__ALT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ( ? [V2l: $i] :
( ( mem @ V2l @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
& ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4N: tp__ty_2Enum_2Enum] :
! [V5n: tp__ty_2Enum_2Enum,V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4N ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
& ( p @ ( ap @ V0P @ V6x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V6x ) ) @ ( ap @ V2l @ V6x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) )
<=> ! [V7e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7e ) ) )
=> ? [V8N: tp__ty_2Enum_2Enum] :
! [V9m: tp__ty_2Enum_2Enum,V10n: tp__ty_2Enum_2Enum,V11x: $i] :
( ( mem @ V11x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V8N ) ) @ ( inj__ty_2Enum_2Enum @ V9m ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V8N ) ) @ ( inj__ty_2Enum_2Enum @ V10n ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V9m ) ) @ ( inj__ty_2Enum_2Enum @ V10n ) ) )
& ( p @ ( ap @ V0P @ V11x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V9m ) ) @ V11x ) ) @ ( ap @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V10n ) ) @ V11x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CAUCHY__IMP__UNIFORMLY__CONVERGENT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4N: tp__ty_2Enum_2Enum] :
! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum,V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4N ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4N ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
& ( p @ ( ap @ V0P @ V7x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ V7x ) ) @ ( ap @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ V7x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V8x ) )
=> ! [V9e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) )
=> ? [V10N: tp__ty_2Enum_2Enum] :
! [V11n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V10N ) ) @ ( inj__ty_2Enum_2Enum @ V11n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V11n ) ) @ V8x ) ) @ ( ap @ V2l @ V8x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) ) ) ) ) ) )
=> ! [V12e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V12e ) ) )
=> ? [V13N: tp__ty_2Enum_2Enum] :
! [V14n: tp__ty_2Enum_2Enum,V15x: $i] :
( ( mem @ V15x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V13N ) ) @ ( inj__ty_2Enum_2Enum @ V14n ) ) )
& ( p @ ( ap @ V0P @ V15x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V14n ) ) @ V15x ) ) @ ( ap @ V2l @ V15x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V12e ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Econtinuous,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1net: $i] :
( ( mem @ V1net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V0f ) @ V1net ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V0f ) @ ( ap @ V0f @ ( ap @ ( c_2Ereal__topology_2Enetlimit @ A_27a ) @ V1net ) ) ) @ V1net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__TRIVIAL__LIMIT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1net: $i] :
( ( mem @ V1net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V1net ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V0f ) @ V1net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V1f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V0s ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V1f ) @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__WITHIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__CLOSED__NONTRIVIAL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3x: tp__ty_2Erealax_2Ereal,V4d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V2s ) )
& ! [V5x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V2s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V1g ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__AT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
& ! [V4x_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V1g ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__OPEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ V2s ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V1g ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__OPEN__IN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ V2s ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) ) @ V3t ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V1g ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) ) @ V3t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__SET__IMP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V2s ) ) ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ V2s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2Econtinuous__within,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V2s ) ) )
<=> ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ! [V5x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2Econtinuous__at,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
& ! [V4x_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__BALL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V1s ) ) )
<=> ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) ) @ V1s ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__BALL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__COMPARISON,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V1g ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) @ V2s ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) @ V2s ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Econtinuous__on,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
=> ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ! [V5x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Euniformly__continuous__on,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V1s ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal,V5x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__IMP__CONTINUOUS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__IMP__CONTINUOUS__ON,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EQ__CONTINUOUS__WITHIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EQ__CONTINUOUS__AT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1s ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__SUBSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V1s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2t ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SUBSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2t ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUBSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2t ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INTERIOR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V1g ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SING,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EMPTY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__NO__LIMPT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ~ ? [V2x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__FINITE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTRACTION__IMP__CONTINUOUS__ON,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EISOMETRY__ON__IMP__CONTINUOUS__ON,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V0s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFORALL__POS__MONO__1,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V1d: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) )
=> ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0P @ ( ap @ c_2Erealax_2Einv @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
=> ! [V4e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
=> ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__SEQUENTIALLY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ V1s ) ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V3x @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V3x ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ c_2Ereal__topology_2Esequentially ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3x ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__SEQUENTIALLY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ c_2Ereal__topology_2Esequentially ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2x ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SEQUENTIALLY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ V1s ) )
& ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V2x @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ c_2Ereal__topology_2Esequentially ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2x ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SEQUENTIALLY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V1s ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V2x @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ V1s ) )
& ! [V5n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V3y @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) @ V1s ) )
& ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V2x @ V6n ) ) @ ( ap @ V3y @ V6n ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ c_2Ereal__topology_2Esequentially ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( ap @ V2x @ V7n ) ) ) @ ( ap @ V0f @ ( ap @ V3y @ V7n ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__CONTINUOUS__FUNCTION,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1net: $i] :
( ( mem @ V1net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V1net ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ V0f @ ( ap @ V2g @ V4x ) ) ) )
@ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) )
@ V1net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CONST,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( inj__ty_2Erealax_2Ereal @ V1c ) ) )
@ V0net ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CMUL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2net: $i] :
( ( mem @ V2net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V0f ) @ V2net ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ V0f @ V3x ) ) ) )
@ V2net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__NEG,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1net: $i] :
( ( mem @ V1net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V0f ) @ V1net ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V2x ) ) ) )
@ V1net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ADD,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2net: $i] :
( ( mem @ V2net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V0f ) @ V2net ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V1g ) @ V2net ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__SUB,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2net: $i] :
( ( mem @ V2net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V0f ) @ V2net ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V1g ) @ V2net ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ABS,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1net: $i] :
( ( mem @ V1net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V0f ) @ V1net ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V2x ) ) ) )
@ V1net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__MAX,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2net: $i] :
( ( mem @ V2net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V0f ) @ V2net ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V1g ) @ V2net ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Emax @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__MIN,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2net: $i] :
( ( mem @ V2net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V0f ) @ V2net ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V1g ) @ V2net ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Emin @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__SUM,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s ) )
& ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3a ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ ( ap @ V1f @ V3a ) ) @ V0net ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] :
( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V2s )
@ ( lam @ A_27b
@ ^ [V5a: $i] : ( ap @ ( ap @ V1f @ V5a ) @ V4x ) ) ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CONST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( inj__ty_2Erealax_2Ereal @ V1c ) ) )
@ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CMUL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ V0f @ V3x ) ) ) )
@ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__NEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V2x ) ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ADD,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V2s ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SUB,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V2s ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ABS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V2x ) ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__MAX,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V2s ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Emax @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__MIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V2s ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Emin @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SUM,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3a ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ ( ap @ V1f @ V3a ) ) @ V0t ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] :
( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V5a: $i] : ( ap @ ( ap @ V1f @ V5a ) @ V4x ) ) ) ) )
@ V0t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CONST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( inj__ty_2Erealax_2Ereal @ V1c ) ) )
@ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__UNIFORMLY__CONTINUOUS__ON,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V1g ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__UNIFORMLY__CONTINUOUS__ON__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V3s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V1g ) @ V3s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V2h ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3s ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3s ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ V2h @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) )
@ V3s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__MUL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V1g ) @ V2s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V2s ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CMUL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V2s ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ V0f @ V3x ) ) ) )
@ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__VMUL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2v: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V1c ) @ V0s ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2v ) ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__NEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V1s ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V2x ) ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__ADD,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V1g ) @ V2s ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUB,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V1g ) @ V2s ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUM,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3a ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ ( ap @ V1f @ V3a ) ) @ V0t ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] :
( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V5a: $i] : ( ap @ ( ap @ V1f @ V5a ) @ V4x ) ) ) ) )
@ V0t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__ID,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : V2x ) )
@ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__ID,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : V1x ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ID,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : V1x ) )
@ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__ID,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : V1x ) )
@ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V3s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V1g ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3s ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V3s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V1g ) @ ( ap @ c_2Ereal__topology_2Eat @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__OPEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V2u ) ) )
<=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V3t ) ) )
=> ? [V4s: $i] :
( ( mem @ V4s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V4s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V4s ) )
& ! [V5x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ V4s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) ) @ V3t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__OPEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
<=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V2t ) ) )
=> ? [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V3s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3s ) )
& ! [V4x_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) @ V3s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) ) @ V2t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN__GEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3u ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) @ V2t ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE__GEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3u ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3u ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__IMP__OPEN__IN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) @ V2t ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSED__GEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3u ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSED,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) @ V2t ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__GEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3u ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3u ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__IMP__CLOSED__IN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) @ V2t ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2t ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__PREIMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2t ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__PREIMAGE__UNIV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V2x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1s ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V1s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE__UNIV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V2x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V1s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2t ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__LIM__0,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__AT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__WITHIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__ON,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__COMPOSE,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V1f ) @ V0net ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V2g ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ V2g @ ( ap @ V1f @ V3x ) ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__ON__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1g ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ V1g @ ( ap @ V0f @ V3x ) ) ) )
@ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__COMPONENT__COMPOSE,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2i: $i] :
( ( mem @ V2i @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V1f ) @ V0net ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ V1f @ V3x ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENT__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ V0f @ V2x ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__CONTINUOUS__COMPOSE,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3h: $i] :
( ( mem @ V3h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V1f ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V2g ) @ V0net ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V3h ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ V3h @ ( ap @ V1f @ V4x ) ) @ ( ap @ V2g @ V4x ) ) ) )
@ V0net ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__CONTINUOUS__ON__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V3s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V2h ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ V2h @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) )
@ V3s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBILINEAR__DOT,axiom,
( p
@ ( ap @ c_2Ereal__topology_2Ebilinear
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V1y: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ V0x ) @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DOT2,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V1f ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V2g ) @ V0net ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2g @ V3x ) ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__DOT2,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V2s ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CONTINUOUS__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__TRANSLATION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V2x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__TRANSLATION__EQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) )
@ V1s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CONTINUOUS__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__TRANSLATION,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__TRANSLATION__EQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) )
@ V1s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EQUASICOMPACT__OPEN__CLOSED,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
=> ( ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3u ) @ V2t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3u ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3u ) ) ) ) )
<=> ! [V5u: $i] :
( ( mem @ V5u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5u ) @ V2t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V5u ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V5u ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__IMP__CONTINUOUS__OPEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
& ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3u ) @ V2t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3u ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__IMP__CONTINUOUS__CLOSED,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
& ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3u ) @ V2t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3u ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__IMP__QUOTIENT__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2t ) ) ) ) ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3t ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3t ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) @ V3t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__IMP__QUOTIENT__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2t ) ) ) ) ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3t ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3t ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) @ V3t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__RIGHT__INVERSE__IMP__QUOTIENT__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3t ) ) @ V2s ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V3t ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) )
= V4y ) ) )
=> ! [V5u: $i] :
( ( mem @ V5u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5u ) @ V3t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V5u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ V5u ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__LEFT__INVERSE__IMP__QUOTIENT__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
= V3x ) ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4u ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ V4u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) ) @ V4u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__OPEN__CLOSED,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
=> ( ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3u ) @ V2t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3u ) ) ) ) )
<=> ! [V5u: $i] :
( ( mem @ V5u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5u ) @ V2t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V5u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V5u ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPOSE__QUOTIENT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3t ) ) @ V4u ) )
& ! [V5v: $i] :
( ( mem @ V5v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5v ) @ V3t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V5v ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ V5v ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) ) @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFUNCTION__FACTORS__LEFT__GEN,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ A_27c ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ V0P @ V3x ) )
& ( p @ ( ap @ V0P @ V4y ) )
& ( ( ap @ V2g @ V3x )
= ( ap @ V2g @ V4y ) ) )
=> ( ( ap @ V1f @ V3x )
= ( ap @ V1f @ V4y ) ) ) ) )
<=> ? [V5h: $i] :
( ( mem @ V5h @ ( arr @ A_27c @ A_27b ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V6x ) )
=> ( ( ap @ V1f @ V6x )
= ( ap @ V5h @ ( ap @ V2g @ V6x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIFT__TO__QUOTIENT__SPACE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s )
= V3t )
& ! [V5v: $i] :
( ( mem @ V5v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5v ) @ V3t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V5v ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ V5v ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1h ) @ V2s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1h ) @ V2s )
= V4u )
& ! [V7x: tp__ty_2Erealax_2Ereal,V8y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8y ) ) @ V2s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V8y ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1h @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1h @ ( inj__ty_2Erealax_2Ereal @ V8y ) ) ) ) ) )
=> ? [V9g: $i] :
( ( mem @ V9g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V9g ) @ V3t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V9g ) @ V3t )
= V4u )
& ! [V10x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V10x ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1h @ ( inj__ty_2Erealax_2Ereal @ V10x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V9g @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V10x ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) @ V3t ) )
& ! [V5v: $i] :
( ( mem @ V5v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5v ) @ V3t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V5v ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ V5v ) ) ) ) )
& ! [V7v: $i] :
( ( mem @ V7v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V7v ) @ V4u ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1g @ V8x ) ) @ V7v ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V4u ) ) @ V7v ) ) ) ) ) )
=> ! [V9v: $i] :
( ( mem @ V9v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V9v ) @ V4u ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V10x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V10x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V10x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) @ V10x ) ) @ V9v ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V4u ) ) @ V9v ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__FROM__COMPOSITION,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3t ) ) @ V4u ) )
& ! [V5v: $i] :
( ( mem @ V5v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5v ) @ V4u ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) @ V6x ) ) @ V5v ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V4u ) ) @ V5v ) ) ) ) ) )
=> ! [V7v: $i] :
( ( mem @ V7v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V7v ) @ V4u ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1g @ V8x ) ) @ V7v ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V4u ) ) @ V7v ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__FROM__SUBSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2t ) ) @ V3u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1s ) @ V2t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V3u )
& ! [V4v: $i] :
( ( mem @ V4v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4v ) @ V3u ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ V4v ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3u ) ) @ V4v ) ) ) ) ) )
=> ! [V6v: $i] :
( ( mem @ V6v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V6v ) @ V3u ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ V6v ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3u ) ) @ V6v ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__RESTRICT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
& ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4u ) @ V2t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ V4u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V4u ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3c ) )
| ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3c ) ) ) )
=> ! [V6u: $i] :
( ( mem @ V6u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V6u ) @ V3c ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal )
@ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ V3c ) ) ) ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x )
@ ( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V9x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V9x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V9x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V9x ) ) @ V3c ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V8x ) ) @ V6u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3c ) ) @ V6u ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__MONOTONE__QUOTIENT__PREIMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3u ) @ V2t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3u ) ) ) ) )
& ! [V5y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V2t ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) ) ) ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V2t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__MONOTONE__QUOTIENT__PREIMAGE__GEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4u ) @ V2t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ V4u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V4u ) ) ) ) )
& ! [V6y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) @ V2t ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) ) ) ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3c ) )
| ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3c ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V3c ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V8x ) ) @ V3c ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__CLOSURES,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) )
<=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__INTERIORS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) )
<=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Einterior @ V2s ) ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__RESTRICT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V4u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V4u ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3t_27 ) @ V2t ) ) )
=> ! [V5u: $i] :
( ( mem @ V5u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal )
@ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V3t_27 ) ) ) ) ) ) )
@ V5u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t_27 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5u ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__RESTRICT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V4u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V4u ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3t_27 ) @ V2t ) ) )
=> ! [V5u: $i] :
( ( mem @ V5u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal )
@ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V3t_27 ) ) ) ) ) ) )
@ V5u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t_27 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5u ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__OPEN__MAP__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
& ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3u ) @ V2t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3u ) ) ) ) ) )
=> ( ! [V5k: $i] :
( ( mem @ V5k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V5k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5k ) ) ) ) )
<=> ! [V6k: $i] :
( ( mem @ V6k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V6k ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V6k ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__CLOSED__MAP__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
& ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3u ) @ V2t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3u ) ) ) ) ) )
=> ( ! [V5k: $i] :
( ( mem @ V5k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V5k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5k ) ) ) ) )
<=> ! [V6k: $i] :
( ( mem @ V6k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V6k ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V6k ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__IMP__OPEN__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3u ) ) ) ) )
& ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V4u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V4u ) ) ) ) ) ) ) ) ) ) )
=> ! [V6u: $i] :
( ( mem @ V6u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V6u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V6u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__IMP__CLOSED__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3u ) ) ) ) )
& ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V4u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V4u ) ) ) ) ) ) ) ) ) ) )
=> ! [V6u: $i] :
( ( mem @ V6u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V6u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V6u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__FROM__COMPOSITION__SURJECTIVE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s )
= V3t )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3t ) ) @ V4u ) )
& ! [V5k: $i] :
( ( mem @ V5k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V5k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V4u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) ) @ V5k ) ) ) ) ) )
=> ! [V6k: $i] :
( ( mem @ V6k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ V6k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V4u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V6k ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__FROM__COMPOSITION__SURJECTIVE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s )
= V3t )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3t ) ) @ V4u ) )
& ! [V5k: $i] :
( ( mem @ V5k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V5k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V4u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) ) @ V5k ) ) ) ) ) )
=> ! [V6k: $i] :
( ( mem @ V6k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ V6k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V4u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V6k ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__FROM__COMPOSITION__INJECTIVE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3t ) ) @ V4u ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3t ) )
& ! [V5x: tp__ty_2Erealax_2Ereal,V6y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) @ V3t ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) ) ) )
=> ( V5x = V6y ) )
& ! [V7k: $i] :
( ( mem @ V7k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V7k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V4u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) ) @ V7k ) ) ) ) ) )
=> ! [V8k: $i] :
( ( mem @ V8k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V8k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V8k ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__FROM__COMPOSITION__INJECTIVE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3t ) ) @ V4u ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3t ) )
& ! [V5x: tp__ty_2Erealax_2Ereal,V6y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) @ V3t ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) ) ) )
=> ( V5x = V6y ) )
& ! [V7k: $i] :
( ( mem @ V7k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V7k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V4u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) ) @ V7k ) ) ) ) ) )
=> ! [V8k: $i] :
( ( mem @ V8k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V8k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V8k ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__CLOSED__SUPERSET__PREIMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4w: $i] :
( ( mem @ V4w @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V5k: $i] :
( ( mem @ V5k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V5k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5k ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4w ) @ V2t ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V4w ) ) ) ) ) )
@ V3u ) ) )
=> ? [V7v: $i] :
( ( mem @ V7v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V7v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4w ) @ V7v ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V8x ) ) @ V7v ) ) ) ) ) )
@ V3u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__CLOSED__SUPERSET__PREIMAGE__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
=> ( ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3k ) ) ) ) )
<=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V5w: $i] :
( ( mem @ V5w @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V4u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5w ) @ V2t ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V5w ) ) ) ) ) )
@ V4u ) ) )
=> ? [V7v: $i] :
( ( mem @ V7v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V7v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5w ) @ V7v ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V8x ) ) @ V7v ) ) ) ) ) )
@ V4u ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4w: $i] :
( ( mem @ V4w @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V5k: $i] :
( ( mem @ V5k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V5k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5k ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4w ) @ V2t ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V4w ) ) ) ) ) )
@ V3u ) ) )
=> ? [V7v: $i] :
( ( mem @ V7v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V7v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4w ) @ V7v ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V8x ) ) @ V7v ) ) ) ) ) )
@ V3u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
=> ( ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3k ) ) ) ) )
<=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V5w: $i] :
( ( mem @ V5w @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V4u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5w ) @ V2t ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V5w ) ) ) ) ) )
@ V4u ) ) )
=> ? [V7v: $i] :
( ( mem @ V7v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V7v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5w ) @ V7v ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V8x ) ) @ V7v ) ) ) ) ) )
@ V4u ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBIGUNION__GSPEC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,A_27h: del,A_27i: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27b @ bool ) @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ bool ) @ ( ap @ V1f @ V2x ) ) @ ( ap @ V0P @ V2x ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V3a: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V3a )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V4x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3a ) @ ( ap @ V1f @ V4x ) ) ) ) ) ) ) ) ) ) )
& ! [V5P: $i] :
( ( mem @ V5P @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27c @ ( arr @ A_27d @ ( arr @ A_27e @ bool ) ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27e )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27e @ bool ) @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ ( ty_2Epair_2Eprod @ ( arr @ A_27e @ bool ) @ bool ) )
@ ( lam @ A_27c
@ ^ [V7x: $i] :
( lam @ A_27d
@ ^ [V8y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27e @ bool ) @ bool ) @ ( ap @ ( ap @ V6f @ V7x ) @ V8y ) ) @ ( ap @ ( ap @ V5P @ V7x ) @ V8y ) ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27e @ A_27e )
@ ( lam @ A_27e
@ ^ [V9a: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27e @ bool ) @ V9a )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27c )
@ ( lam @ A_27c
@ ^ [V10x: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27d )
@ ( lam @ A_27d
@ ^ [V11y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V5P @ V10x ) @ V11y ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27e ) @ V9a ) @ ( ap @ ( ap @ V6f @ V10x ) @ V11y ) ) ) ) ) ) ) ) ) ) ) ) )
& ! [V12P: $i] :
( ( mem @ V12P @ ( arr @ A_27f @ ( arr @ A_27g @ ( arr @ A_27h @ bool ) ) ) )
=> ! [V13f: $i] :
( ( mem @ V13f @ ( arr @ A_27f @ ( arr @ A_27g @ ( arr @ A_27h @ ( arr @ A_27i @ bool ) ) ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27i )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27i @ bool ) @ ( ty_2Epair_2Eprod @ A_27f @ ( ty_2Epair_2Eprod @ A_27g @ A_27h ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27f @ ( ty_2Epair_2Eprod @ A_27g @ A_27h ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27i @ bool ) @ bool ) )
@ ( lam @ A_27f
@ ^ [V14x: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27g @ A_27h @ ( ty_2Epair_2Eprod @ ( arr @ A_27i @ bool ) @ bool ) )
@ ( lam @ A_27g
@ ^ [V15y: $i] :
( lam @ A_27h
@ ^ [V16z: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27i @ bool ) @ bool ) @ ( ap @ ( ap @ ( ap @ V13f @ V14x ) @ V15y ) @ V16z ) ) @ ( ap @ ( ap @ ( ap @ V12P @ V14x ) @ V15y ) @ V16z ) ) ) ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27i @ A_27i )
@ ( lam @ A_27i
@ ^ [V17a: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27i @ bool ) @ V17a )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27f )
@ ( lam @ A_27f
@ ^ [V18x: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27g )
@ ( lam @ A_27g
@ ^ [V19y: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27h )
@ ( lam @ A_27h
@ ^ [V20z: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( ap @ V12P @ V18x ) @ V19y ) @ V20z ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27i ) @ V17a ) @ ( ap @ ( ap @ ( ap @ V13f @ V18x ) @ V19y ) @ V20z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE__POINT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
=> ( ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3k ) ) ) ) )
<=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V5y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V4u ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V2t ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) ) ) ) )
@ V4u ) ) )
=> ? [V7v: $i] :
( ( mem @ V7v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V7v ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V7v ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V8x ) ) @ V7v ) ) ) ) ) )
@ V4u ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__MONOTONE__PREIMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V3c: $i] :
( ( mem @ V3c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3c ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3c ) ) ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V2t ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) ) ) ) ) ) )
=> ! [V6c: $i] :
( ( mem @ V6c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V6c ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V6c ) @ V2t ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ V6c ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__MONOTONE__PREIMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V3c: $i] :
( ( mem @ V3c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3c ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3c ) ) ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V2t ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) ) ) ) ) ) )
=> ! [V6c: $i] :
( ( mem @ V6c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V6c ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V6c ) @ V2t ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ V6c ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPROPER__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
=> ( ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3k ) @ V2t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V3k ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3k ) ) ) ) ) ) ) ) )
<=> ( ! [V5k: $i] :
( ( mem @ V5k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V5k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5k ) ) ) ) )
& ! [V6a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ V2t ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CONTINUOUS__IMAGE__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) )
=> ( V2x = V3y ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V4t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4t ) @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V4t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPACT,axiom,
! [V0t: $i] :
( ( mem @ V0t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1f ) @ V2s ) ) @ V0t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0t ) ) @ V3k ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V4x ) ) @ V3k ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPROPER__MAP__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) @ V3t ) )
& ! [V5k: $i] :
( ( mem @ V5k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5k ) @ V3t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V5k ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V5k ) ) ) ) ) ) ) ) )
& ! [V7k: $i] :
( ( mem @ V7k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V7k ) @ V4u ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V7k ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1g @ V8x ) ) @ V7k ) ) ) ) ) ) ) ) ) )
=> ! [V9k: $i] :
( ( mem @ V9k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V9k ) @ V4u ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V9k ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V10x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V10x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V10x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) @ V10x ) ) @ V9k ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPOSITION__LEFT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s )
= V3t )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3t ) ) @ V4u ) )
& ! [V5k: $i] :
( ( mem @ V5k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5k ) @ V4u ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V5k ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) @ V6x ) ) @ V5k ) ) ) ) ) ) ) ) ) )
=> ! [V7k: $i] :
( ( mem @ V7k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V7k ) @ V4u ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V7k ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1g @ V8x ) ) @ V7k ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPOSITION__RIGHT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3t ) ) @ V4u ) )
& ! [V5k: $i] :
( ( mem @ V5k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5k ) @ V4u ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V5k ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) @ V6x ) ) @ V5k ) ) ) ) ) ) ) ) ) )
=> ! [V7k: $i] :
( ( mem @ V7k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V7k ) @ V3t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V7k ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V8x ) ) @ V7k ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPASTING__LEMMA,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4k: $i] :
( ( mem @ V4k @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V5i: $i] :
( ( mem @ V5i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5i ) @ V4k ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3s ) ) @ ( ap @ V2t @ V5i ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ ( ap @ V0f @ V5i ) ) @ ( ap @ V2t @ V5i ) ) ) ) ) )
& ! [V6i: $i] :
( ( mem @ V6i @ A_27a )
=> ! [V7j: $i] :
( ( mem @ V7j @ A_27a )
=> ! [V8x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6i ) @ V4k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7j ) @ V4k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V3s ) @ ( ap @ V2t @ V6i ) ) ) @ ( ap @ V2t @ V7j ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0f @ V6i ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0f @ V7j ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) ) ) ) )
& ! [V9x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) @ V3s ) )
=> ? [V10j: $i] :
( ( mem @ V10j @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V10j ) @ V4k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) @ ( ap @ V2t @ V10j ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0f @ V10j ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPASTING__LEMMA__EXISTS,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ A_27a @ bool ) )
=> ( ( ( p
@ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2s )
@ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ A_27a )
@ ( lam @ A_27a
@ ^ [V4i: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( ap @ V1t @ V4i ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4i ) @ V3k ) ) ) ) ) ) )
& ! [V5i: $i] :
( ( mem @ V5i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5i ) @ V3k ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ ( ap @ V1t @ V5i ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ ( ap @ V0f @ V5i ) ) @ ( ap @ V1t @ V5i ) ) ) ) ) )
& ! [V6i: $i] :
( ( mem @ V6i @ A_27a )
=> ! [V7j: $i] :
( ( mem @ V7j @ A_27a )
=> ! [V8x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6i ) @ V3k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7j ) @ V3k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2s ) @ ( ap @ V1t @ V6i ) ) ) @ ( ap @ V1t @ V7j ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0f @ V6i ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0f @ V7j ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) ) ) ) ) )
=> ? [V9g: $i] :
( ( mem @ V9g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V9g ) @ V2s ) )
& ! [V10x: tp__ty_2Erealax_2Ereal,V11i: $i] :
( ( mem @ V11i @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V11i ) @ V3k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V10x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2s ) @ ( ap @ V1t @ V11i ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V9g @ ( inj__ty_2Erealax_2Ereal @ V10x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0f @ V11i ) @ ( inj__ty_2Erealax_2Ereal @ V10x ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__LOCAL__OPEN,axiom,
! [V0t: $i] :
( ( mem @ V0t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V2s ) @ V0t ) ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V2s ) @ V0t ) ) ) @ V0t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V0t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V2s ) @ V0t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__OPEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LOCAL__OPEN,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) ) ) @ V3s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) ) ) @ V4t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V3s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V2g ) @ V4t ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V3s ) )
& ~ ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) )
| ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V4t ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ V0P @ V6x ) ) @ ( ap @ V1f @ V6x ) ) @ ( ap @ V2g @ V6x ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__OPEN,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V3s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V4t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V3s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V2g ) @ V4t ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V3s ) )
& ~ ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) )
| ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V4t ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ V0P @ V6x ) ) @ ( ap @ V1f @ V6x ) ) @ ( ap @ V2g @ V6x ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPASTING__LEMMA__CLOSED,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4k: $i] :
( ( mem @ V4k @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V4k ) )
& ! [V5i: $i] :
( ( mem @ V5i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5i ) @ V4k ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3s ) ) @ ( ap @ V2t @ V5i ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ ( ap @ V0f @ V5i ) ) @ ( ap @ V2t @ V5i ) ) ) ) ) )
& ! [V6i: $i] :
( ( mem @ V6i @ A_27a )
=> ! [V7j: $i] :
( ( mem @ V7j @ A_27a )
=> ! [V8x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6i ) @ V4k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7j ) @ V4k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V3s ) @ ( ap @ V2t @ V6i ) ) ) @ ( ap @ V2t @ V7j ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0f @ V6i ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0f @ V7j ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) ) ) ) )
& ! [V9x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) @ V3s ) )
=> ? [V10j: $i] :
( ( mem @ V10j @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V10j ) @ V4k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) @ ( ap @ V2t @ V10j ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0f @ V10j ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPASTING__LEMMA__EXISTS__CLOSED,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3k ) )
& ( p
@ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2s )
@ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ A_27a )
@ ( lam @ A_27a
@ ^ [V4i: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( ap @ V1t @ V4i ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4i ) @ V3k ) ) ) ) ) ) )
& ! [V5i: $i] :
( ( mem @ V5i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5i ) @ V3k ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ ( ap @ V1t @ V5i ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ ( ap @ V0f @ V5i ) ) @ ( ap @ V1t @ V5i ) ) ) ) ) )
& ! [V6i: $i] :
( ( mem @ V6i @ A_27a )
=> ! [V7j: $i] :
( ( mem @ V7j @ A_27a )
=> ! [V8x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6i ) @ V3k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7j ) @ V3k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2s ) @ ( ap @ V1t @ V6i ) ) ) @ ( ap @ V1t @ V7j ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0f @ V6i ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0f @ V7j ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) ) ) ) ) )
=> ? [V9g: $i] :
( ( mem @ V9g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V9g ) @ V2s ) )
& ! [V10x: tp__ty_2Erealax_2Ereal,V11i: $i] :
( ( mem @ V11i @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V11i ) @ V3k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V10x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2s ) @ ( ap @ V1t @ V11i ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V9g @ ( inj__ty_2Erealax_2Ereal @ V10x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0f @ V11i ) @ ( inj__ty_2Erealax_2Ereal @ V10x ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__LIFT__DOT,axiom,
! [V0l: tp__ty_2Erealax_2Ereal,V1net: tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ ( inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ V1net ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4y: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( ap @ V2f @ V4y ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) )
@ ( inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ V1net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__LIFT__DOT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2y ) ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__LIFT__DOT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2y ) ) )
@ V1s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL__LEFT,axiom,
! [V0b: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL__RIGHT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__HALFSPACE__LE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__HALFSPACE__GE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__HYPERPLANE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__HYPERPLANE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosure
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__STANDARD__HYPERPLANE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__HALFSPACE__COMPONENT__LE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__HALFSPACE__COMPONENT__GE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__HALFSPACE__LT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__HALFSPACE__COMPONENT__LT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__HALFSPACE__GT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__gt @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__HALFSPACE__COMPONENT__GT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__gt @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__POSITIVE__MULTIPLES,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1c: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ V1c ) @ V2x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1c ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__LEFT,axiom,
! [V0b: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__RIGHT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__POSITIVE__ORTHANT,axiom,
( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V0x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__HALFSPACE__LE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( V0a
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__HALFSPACE__GE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( V0a
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__gt @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__HALFSPACE__COMPONENT__LE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Einterior
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__HALFSPACE__COMPONENT__GE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Einterior
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__gt @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__HALFSPACE__LT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( V0a
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__HALFSPACE__GT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( V0a
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__gt @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__HALFSPACE__COMPONENT__LT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosure
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__HALFSPACE__COMPONENT__GT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosure
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__gt @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__HYPERPLANE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( V0a
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__HALFSPACE__LE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ~ ( ( V0a
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1b
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Efrontier
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__HALFSPACE__GE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ~ ( ( V0a
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1b
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Efrontier
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__HALFSPACE__LT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ~ ( ( V0a
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1b
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Efrontier
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__HALFSPACE__GT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ~ ( ( V0a
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1b
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Efrontier
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__gt @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__STANDARD__HYPERPLANE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Einterior
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__LE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
~ ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__GE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
~ ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__LT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
~ ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__GT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
~ ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__gt @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFORALL__IN__CLOSURE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V2t ) ) ) )
=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFORALL__IN__CLOSURE__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) ) )
=> ( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V2t ) ) )
<=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ V2t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__LE__ON__CLOSURE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) )
=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__GE__ON__CLOSURE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) )
=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CONSTANT__ON__CLOSURE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= V2a ) ) )
=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= V2a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AGREE__ON__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V2h ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2h @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) )
=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2h @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__CONSTANT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE__CONSTANT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V2s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2s ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal,V3e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) )
=> ? [V4d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ! [V5y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__SEQUENTIALLY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V2x @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ c_2Ereal__topology_2Esequentially ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2x ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CLOSURE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__IMP__CAUCHY__CONTINUOUS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V1s ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ V2x ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V2x @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IMP__CAUCHY__CONTINUOUS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ V2x ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V2x @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__UNIQUENESS__LEMMA,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ V2x ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V2x @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2x ) ) ) ) )
=> ! [V4a: tp__ty_2Erealax_2Ereal,V5x: $i] :
( ( mem @ V5x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V6n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V5x @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V5x ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ c_2Ereal__topology_2Esequentially ) ) )
=> ? [V7l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V7l ) ) @ c_2Ereal__topology_2Esequentially ) )
& ! [V8y: $i] :
( ( mem @ V8y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V9n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V8y @ ( inj__ty_2Enum_2Enum @ V9n ) ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V8y ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ c_2Ereal__topology_2Esequentially ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V8y ) ) @ ( inj__ty_2Erealax_2Ereal @ V7l ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__EXTENDS__TO__CLOSURE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ V2x ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V2x @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2x ) ) ) ) )
=> ? [V4g: $i] :
( ( mem @ V4g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V4g ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V4g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__EXTENDS__TO__CLOSURE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V1s ) )
=> ? [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V2g ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
& ! [V4h: $i] :
( ( mem @ V4h @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V4h ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V4h @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) ) )
=> ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V4h @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__IMP__CONTINUOUS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ V2x ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V2x @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__UNIFORMLY__CONTINUOUS__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__COMPOSE__EQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3h: $i] :
( ( mem @ V3h @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V2g ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V3h ) @ ( ap @ c_2Ereal__topology_2Eat @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( ap @ V3h @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) )
= V4y )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V3h @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= V0x ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V1f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ V1f @ ( ap @ V2g @ V5x ) ) ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__TRANSLATION,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1z: tp__ty_2Erealax_2Ereal,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V2f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) ) ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ V2f @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__IMAGE__SUBSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V2x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) )
=> ( V3x = V4y ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__AVOID,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V2s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
!= V3a ) )
=> ? [V4e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
& ! [V5y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) )
!= V3a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__AVOID,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
!= V2a ) )
=> ? [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) )
!= V2a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__AVOID,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
!= V3a ) )
=> ? [V4e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
& ! [V5y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) )
!= V3a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN__AVOID,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
!= V3a ) )
=> ? [V4e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
& ! [V5y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) )
!= V3a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN__IN__CASES,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ) ) )
=> ( ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
!= V2a ) )
| ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= V2a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN__IN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ) )
& ? [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= V2a ) ) )
=> ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= V2a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ) )
& ? [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= V2a ) ) )
=> ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= V2a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__SCALING,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( ( V1c
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ V2x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__NEGATIONS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V1x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__TRANSLATION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V2x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__TRANSLATION__EQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) )
@ V1s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__AFFINITY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
& ( V2c
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ V3x ) ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__TRANSLATION,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) )
@ V1s ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) )
@ ( ap @ c_2Ereal__topology_2Einterior @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__SUMS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
| ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EOpen
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUPPER__HEMICONTINUOUS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V1t ) ) )
=> ( ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V4u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ V4u ) ) ) ) ) ) ) ) )
<=> ! [V6u: $i] :
( ( mem @ V6u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V6u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V2s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ V6u ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOWER__HEMICONTINUOUS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V1t ) ) )
=> ( ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V4u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ V4u ) ) ) ) ) ) ) ) )
<=> ! [V6u: $i] :
( ( mem @ V6u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V6u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V2s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ V6u ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__IFF__LOWER__HEMICONTINUOUS__PREIMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
=> ( ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3u ) ) ) ) )
<=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V4u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5y )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5y ) @ V2t ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V5y ) ) ) ) ) )
@ V4u ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__IFF__UPPER__HEMICONTINUOUS__PREIMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
=> ( ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3u ) ) ) ) )
<=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V4u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5y )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5y ) @ V2t ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V5y ) ) ) ) ) )
@ V4u ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUPPER__LOWER__HEMICONTINUOUS__EXPLICIT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V1t ) ) )
& ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V4u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ V4u ) ) ) ) ) ) ) ) )
& ! [V6u: $i] :
( ( mem @ V6u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V6u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ V6u ) ) ) ) ) ) ) ) ) )
=> ! [V8x: tp__ty_2Erealax_2Ereal,V9e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) )
& ( ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V8x ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V10d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V10d ) ) )
& ! [V11x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V11x_27 ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ ( inj__ty_2Erealax_2Ereal @ V11x_27 ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V10d ) ) ) )
=> ( ! [V12y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12y ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) )
=> ? [V13y_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V13y_27 ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V11x_27 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12y ) ) @ ( inj__ty_2Erealax_2Ereal @ V13y_27 ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) ) ) )
& ! [V14y_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V14y_27 ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V11x_27 ) ) ) )
=> ? [V15y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V15y ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V14y_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V15y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Econnected__component,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
<=> ? [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3t ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V3t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__IN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__REFL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__REFL__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SYM,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__TRANS,axiom,
! [V0z: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__OF__SUBSET,axiom,
! [V0y: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1s ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V2t ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2y )
@ ( ap @ ( c_2Ebool_2E_3F @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ( arr @ ty_2Erealax_2Ereal @ bool )
@ ^ [V3t: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ereal__topology_2Econnected @ V3t ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3t ) @ V0s ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2y ) @ V3t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__BIGUNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ( arr @ ty_2Erealax_2Ereal @ bool )
@ ^ [V2t: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ V2t ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ereal__topology_2Econnected @ V2t ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2t ) @ V0s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CONNECTED__COMPONENT__SET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) )
= V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__UNIV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) )
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__UNIV,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) )
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) )
<=> ( V0s
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__SELF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) )
=> ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) )
= V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__IFF__CONNECTED__COMPONENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__MAXIMAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__MONO,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V1t ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CONNECTED__COMPONENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__EMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EMPTY,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
=> ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) )
= ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__CONNECTED__COMPONENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__DISJOINT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__NONOVERLAP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) )
| ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ V0s ) )
| ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) )
!= ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__OVERLAP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ V0s ) )
& ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) )
= ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SYM__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) )
= ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
<=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) ) )
| ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EQ__CONNECTED__COMPONENT__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) ) )
=> ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) )
= ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__IDEMP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) )
= ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__UNIQUE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1c ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1c ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1c ) )
& ! [V3c_27: $i] :
( ( mem @ V3c_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V3c_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3c_27 ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V3c_27 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3c_27 ) @ V1c ) ) ) ) )
=> ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) )
= V1c ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EJOINABLE__CONNECTED__COMPONENT__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) )
= ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBIGUNION__CONNECTED__COMPONENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ V1x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V1x ) @ V0s ) ) ) ) )
= V0s ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPLEMENT__CONNECTED__COMPONENT__BIGUNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EDELETE @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ V2y ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2y ) @ V0s ) ) ) ) )
@ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__CONNECTED__COMPONENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBIGUNION__DIFF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0s ) ) @ V1t )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V2x ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2x ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__CONNECTED__COMPONENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQUIVALENCE__RELATION,axiom,
! [V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal,V5y: tp__ty_2Erealax_2Ereal,V6z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) )
& ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V6z ) ) ) )
& ! [V7a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7a ) ) @ V1s ) )
=> ? [V8t: $i] :
( ( mem @ V8t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V8t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7a ) ) @ V8t ) )
& ! [V9x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) @ V8t ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V7a ) ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) ) ) ) ) )
=> ! [V10a: tp__ty_2Erealax_2Ereal,V11b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V10a ) ) @ ( inj__ty_2Erealax_2Ereal @ V11b ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( inj__ty_2Erealax_2Ereal @ V10a ) ) @ ( inj__ty_2Erealax_2Ereal @ V11b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__INTERMEDIATE__SUBSET,axiom,
! [V0t: $i] :
( ( mem @ V0t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V1u ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ V0t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0t ) @ V1u ) ) )
=> ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0t ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) )
= ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V1u ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ecomponents,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Ecomponents @ V0s )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0s ) @ V1x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V1x ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0u ) ) )
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0u ) )
& ( V1s
= ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0u ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBIGUNION__COMPONENTS,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( V0u
= ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0u ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPAIRWISE__DISJOINT__COMPONENTS,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epairwise @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( c_2Epred__set_2EDISJOINT @ ty_2Erealax_2Ereal ) ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0u ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS__NONEMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
=> ( V1c
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1c ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS__CONNECTED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1c ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS__MAXIMAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
<=> ( ( V1c
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1c ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1c ) )
& ! [V2c_27: $i] :
( ( mem @ V2c_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( V2c_27
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1c ) @ V2c_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2c_27 ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V2c_27 ) ) )
=> ( V2c_27 = V1c ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EJOINABLE__COMPONENTS__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2c1: $i] :
( ( mem @ V2c1 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3c2: $i] :
( ( mem @ V3c2 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2c1 ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3c2 ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2c1 ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V3c2 ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( V2c1 = V3c2 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__COMPONENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1c ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__COMPONENTS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1c ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__COMPONENTS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1c ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENTS__GEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V1s ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V2c ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2c ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENTS__FINITE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V1s ) ) )
& ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2c ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__NONOVERLAP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2c_27: $i] :
( ( mem @ V2c_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2c_27 ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1c ) @ V2c_27 )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( V1c != V2c_27 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2c_27: $i] :
( ( mem @ V2c_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2c_27 ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) ) )
=> ( ( V1c = V2c_27 )
<=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1c ) @ V2c_27 )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__EMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Ecomponents @ V0s )
= ( c_2Epred__set_2EEMPTY @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__EMPTY,axiom,
( ( ap @ c_2Ereal__topology_2Ecomponents @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
= ( c_2Epred__set_2EEMPTY @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EQ__CONNECTED__COMPONENTS__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2c_27: $i] :
( ( mem @ V2c_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2c_27 ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) ) )
=> ( V1c = V2c_27 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__SING__N__EXISTS,axiom,
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Ecomponents @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ) )
<=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ? [V2a: $i] :
( ( mem @ V2a @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( ( ap @ c_2Ereal__topology_2Ecomponents @ V1s )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2a ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ) ) )
<=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__SING,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Ecomponents @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ) )
<=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__SING__EXISTS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ? [V1a: $i] :
( ( mem @ V1a @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( ( ap @ c_2Ereal__topology_2Ecomponents @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1a ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ) ) )
<=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__UNIV,axiom,
( ( ap @ c_2Ereal__topology_2Ecomponents @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EQ__COMPONENTS__SUBSET__SING,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EQ__COMPONENTS__SUBSET__SING__EXISTS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ? [V1a: $i] :
( ( mem @ V1a @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1a ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS__SELF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
<=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__MAXIMAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2c ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V2c ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__UNIQUE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V1k )
= V0s )
& ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2c ) @ V1k ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V2c ) )
& ( V2c
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V3c_27: $i] :
( ( mem @ V3c_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V3c_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2c ) @ V3c_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3c_27 ) @ V0s ) ) )
=> ( V3c_27 = V2c ) ) ) ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Ecomponents @ V0s )
= V1k ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__UNIQUE__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( ap @ c_2Ereal__topology_2Ecomponents @ V0s )
= V1k )
<=> ( ( ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V1k )
= V0s )
& ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2c ) @ V1k ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V2c ) )
& ( V2c
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V3c_27: $i] :
( ( mem @ V3c_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V3c_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2c ) @ V3c_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3c_27 ) @ V0s ) ) )
=> ( V3c_27 = V2c ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEXISTS__COMPONENT__SUPERSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1t ) ) )
=> ? [V2c: $i] :
( ( mem @ V2c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V2c ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__INTERMEDIATE__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V2u ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS__BIGUNION__COMPLEMENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1c ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V0s ) @ V1c )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0s ) ) @ V1c ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__SUBSET__CLOPEN,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V2c ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2c ) @ V0u ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2c ) @ V1s )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2c ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOPEN__BIGUNION__COMPONENTS,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ V1s ) ) )
=> ? [V2k: $i] :
( ( mem @ V2k @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2k ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0u ) ) )
& ( V1s
= ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V2k ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOPEN__IN__COMPONENTS,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ ( ap @ c_2Ereal__topology_2Ecomponents @ V0u ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__UNIFORMLY__EQUICONTINUOUS,axiom,
! [V0fs: $i] :
( ( mem @ V0fs @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V2x: tp__ty_2Erealax_2Ereal,V3e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) )
=> ? [V4d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V6x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V5f ) @ V0fs ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x_27 ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V5f @ ( inj__ty_2Erealax_2Ereal @ V6x_27 ) ) ) @ ( ap @ V5f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) ) )
=> ! [V7e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7e ) ) )
=> ? [V8d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V8d ) ) )
& ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V10x: tp__ty_2Erealax_2Ereal,V11x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V9f ) @ V0fs ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V10x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V11x_27 ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V11x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V10x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V8d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V9f @ ( inj__ty_2Erealax_2Ereal @ V11x_27 ) ) ) @ ( ap @ V9f @ ( inj__ty_2Erealax_2Ereal @ V10x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__UNIFORMLY__CONTINUOUS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EABS__TRIANGLE__LE,axiom,
! [V0e: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__UNIFORM__LIMIT,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ ( ap @ V1f @ V4n ) ) @ V3s ) ) )
@ V0net ) )
& ! [V5e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V6n: $i] :
( ap @ ( c_2Ebool_2E_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V3s ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V1f @ V6n ) @ V7x ) ) @ ( ap @ V2g @ V7x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) ) ) ) ) )
@ V0net ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V2g ) @ V3s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ! [V3x_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V3x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x_27 ) ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3x_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x_27 ) ) @ V0s ) )
& ( V3x_27 != V1x )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V3x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__RANGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
& ! [V4x_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__RANGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
=> ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ! [V5x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x_27 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ABS__COMPOSE,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V1f ) @ V0net ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V1f @ V2x ) ) ) )
@ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ABS__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V2x ) ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__ABS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ c_2Ereal_2Eabs ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__DIST,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__DIST,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) ) )
@ V1s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__ATTAINS__SUP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
& ! [V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__ATTAINS__INF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
& ! [V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ATTAINS__SUP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) ) )
=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
& ! [V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ATTAINS__INF,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) ) )
=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
& ! [V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDISTANCE__ATTAINS__SUP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ! [V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDISTANCE__ATTAINS__INF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ! [V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__MUL,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3c: $i] :
( ( mem @ V3c @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V4d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V3c ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V3c @ V5x ) ) @ ( ap @ V1f @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__VMUL,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2d: tp__ty_2Erealax_2Ereal,V3v: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1c ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) @ V0net ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3v ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) @ ( inj__ty_2Erealax_2Ereal @ V3v ) ) )
@ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__VMUL,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2v: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V1c ) @ V0net ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2v ) ) ) )
@ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__MUL,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V2c ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V1f ) @ V0net ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2c @ V3x ) ) @ ( ap @ V1f @ V3x ) ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__VMUL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2v: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1c ) @ V0s ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2v ) ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__MUL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1c ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V2f ) @ V0s ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V3x ) ) @ ( ap @ V2f @ V3x ) ) ) )
@ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__POW,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ V1f @ V3x ) ) )
@ V0net ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ V1f @ V4x ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
@ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__POW,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ V0f @ V3x ) ) )
@ V1s ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ V0f @ V4x ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__PRODUCT,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) )
& ! [V3i: $i] :
( ( mem @ V3i @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3i ) @ V2t ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ V1f @ V4x ) @ V3i ) ) )
@ V0net ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27b ) @ V2t ) @ ( ap @ V1f @ V5x ) ) ) )
@ V0net ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__PRODUCT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) )
& ! [V3i: $i] :
( ( mem @ V3i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3i ) @ V2t ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ V0f @ V4x ) @ V3i ) ) )
@ V1s ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Eproduct_2Eproduct @ A_27a ) @ V2t ) @ ( ap @ V0f @ V5x ) ) ) )
@ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__INV,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) )
& ( V2l
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Einv ) @ V1f ) ) @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) ) @ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__INV,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ V1f ) @ V0net ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( ap @ ( c_2Ereal__topology_2Enetlimit @ A_27a ) @ V0net ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Einv ) @ V1f ) ) @ V0net ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__WITHIN__INV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ V1s ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Einv ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__INV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Einv ) @ V0f ) ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Einv ) @ V0f ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__SCALING,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ V2x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__NEGATIONS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V1x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__SCALING,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ V2x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__NEGATIONS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V1x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__AFFINITY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ V3x ) ) ) )
@ V0s ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ediameter,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ ( ap @ c_2Ereal_2Esup
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ V1x ) @ V2y ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V1x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2y ) @ V0s ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__BOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ( ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) ) ) )
& ! [V3d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) ) ) )
=> ? [V4x: tp__ty_2Erealax_2Ereal,V5y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__gt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__BOUNDED__BOUND,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ediameter @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ediameter @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__EMPTY,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ediameter @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ediameter @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__POS__LE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Ediameter @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ediameter @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__SUBSET__CBALL__NONEMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V1z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__SUBSET__CBALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ? [V1z: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__EQ__0,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ? [V1a: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__LE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1d: tp__ty_2Erealax_2Ereal] :
( ( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__CBALL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ediameter @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__BALL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ediameter @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__SUMS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal_2Ereal__lte
@ ( ap @ c_2Ereal__topology_2Ediameter
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Ediameter @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELEBESGUE__COVERING__LEMMA,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( V1c
!= ( c_2Epred__set_2EEMPTY @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V1c ) ) )
& ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2b ) @ V1c ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2b ) ) ) ) )
=> ? [V3d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
& ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4t ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Ediameter @ V4t ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) )
=> ? [V5b: $i] :
( ( mem @ V5b @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V5b ) @ V1c ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4t ) @ V5b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__SCALING,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ V2x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__NEGATIONS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V1x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CLOSED__SUMS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__COMPACT__SUMS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__SUMS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
| ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V5y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V4x ) @ V5y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5y ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CLOSED__DIFFERENCES,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__COMPACT__DIFFERENCES,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ETRANSLATION__DIFF,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V1s ) @ V2t ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V4x ) ) )
@ V1s ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V5x ) ) )
@ V2t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEPARATE__POINT__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) ) )
=> ? [V2d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEPARATE__COMPACT__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V2d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEPARATE__CLOSED__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V2d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__UNION__COMPACT__SUBSETS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0s ) )
& ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V1f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( ap @ V1f @ V5n ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V5n ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) )
= V0s )
& ! [V6k: $i] :
( ( mem @ V6k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V6k ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V6k ) @ V0s ) ) )
=> ? [V7N: tp__ty_2Enum_2Enum] :
! [V8n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V8n ) ) @ ( inj__ty_2Enum_2Enum @ V7N ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V6k ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__UNION__COMPACT__SUBSETS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0s ) )
& ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ V1f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( ap @ V1f @ V5n ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V5n ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) )
= V0s )
& ! [V6k: $i] :
( ( mem @ V6k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V6k ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V6k ) @ V0s ) ) )
=> ? [V7N: tp__ty_2Enum_2Enum] :
! [V8n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V8n ) ) @ ( inj__ty_2Enum_2Enum @ V7N ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V6k ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2EOPEN__interval,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2ECLOSED__interval,axiom,
! [V0l: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0l ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( c_2Epair_2EFST @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( c_2Elist_2EHD @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0l ) ) ) ) @ V1x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V1x ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( c_2Elist_2EHD @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0l ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2Einterval,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) )
& ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V3x ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__INTERVAL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__INTERVAL__REFLECT,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) )
& ! [V3a: tp__ty_2Erealax_2Ereal,V4b: tp__ty_2Erealax_2Ereal,V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREFLECT__INTERVAL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V2x ) ) )
@ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
& ! [V3a: tp__ty_2Erealax_2Ereal,V4b: tp__ty_2Erealax_2Ereal] :
( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V5x ) ) )
@ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) )
= ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERVAL__EQ__EMPTY,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
<=> ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
<=> ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERVAL__NE__EMPTY,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
& ( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSET__INTERVAL__IMP,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1c: tp__ty_2Erealax_2Ereal,V2d: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) )
& ( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) )
& ( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) )
& ( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERVAL__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
& ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSET__INTERVAL,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1d: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDISJOINT__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) )
& ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) )
& ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) )
& ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
| ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EENDS__IN__INTERVAL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
& ! [V4a: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal] :
~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) ) )
& ! [V6a: tp__ty_2Erealax_2Ereal,V7b: tp__ty_2Erealax_2Ereal] :
~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EENDS__IN__UNIT__INTERVAL,axiom,
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTER__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Emax @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Emin @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERVAL__OPEN__SUBSET__CLOSED,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LT__MIN,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ ( ap @ ( ap @ c_2Ereal_2Emin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__LEMMA,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
=> ? [V3d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
& ! [V4x_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CLOSED__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__INTERVAL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
= ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__INTERVAL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENOT__INTERVAL__UNIV,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
!= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
!= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__MIDPOINT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__CLOSED__INTERVAL__CONVEX,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal,V4e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LE__INV2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__INV__LE__1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__INTERVAL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__OPEN__INTERVAL__SYMMETRIC,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ? [V1a: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__OPEN__INTERVAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ? [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CLOSED__INTERVAL__SYMMETRIC,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ? [V1a: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CLOSED__INTERVAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
=> ? [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__CLOSED__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFRONTIER__OPEN__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Efrontier @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTER__INTERVAL__MIXED__EQ__EMPTY,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
<=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERVAL__TRANSLATION,axiom,
( ! [V0c: tp__ty_2Erealax_2Ereal,V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ V3x ) ) )
@ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ! [V4c: tp__ty_2Erealax_2Ereal,V5a: tp__ty_2Erealax_2Ereal,V6b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V4c ) ) @ ( inj__ty_2Erealax_2Ereal @ V5a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V4c ) ) @ ( inj__ty_2Erealax_2Ereal @ V6b ) ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V4c ) ) @ V7x ) ) )
@ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEMPTY__AS__INTERVAL,axiom,
( ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIT__INTERVAL__NONEMPTY,axiom,
( ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIMAGE__STRETCH__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2m: $i] :
( ( mem @ V2m @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] :
( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4f: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V4f ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2m @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V3x ) ) ) ) ) )
@ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
@ ( ap @ c_2Ereal__topology_2ECLOSED__interval
@ ( ap
@ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap
@ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5f: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V5f ) @ ( ap @ ( ap @ c_2Ereal_2Emin @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2m @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2m @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) )
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6f: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V6f ) @ ( ap @ ( ap @ c_2Ereal_2Emax @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2m @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2m @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) )
@ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERVAL__IMAGE__STRETCH__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2m: $i] :
( ( mem @ V2m @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ? [V3u: tp__ty_2Erealax_2Ereal,V4v: tp__ty_2Erealax_2Ereal] :
( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] :
( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6f: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V6f ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2m @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V5x ) ) ) ) ) )
@ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3u ) ) @ ( inj__ty_2Erealax_2Ereal @ V4v ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL__IMAGE__UNIT__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
=> ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] :
( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4f: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V4f ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ V3x ) ) ) ) ) )
@ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__INTERVALS,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V5y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V4x ) @ V5y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5y ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ) ) )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ! [V6a: tp__ty_2Erealax_2Ereal,V7b: tp__ty_2Erealax_2Ereal,V8c: tp__ty_2Erealax_2Ereal,V9d: tp__ty_2Erealax_2Ereal] :
( ( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8c ) ) @ ( inj__ty_2Erealax_2Ereal @ V9d ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V10x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V11y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V10x ) @ V11y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V10x ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V11y ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8c ) ) @ ( inj__ty_2Erealax_2Ereal @ V9d ) ) ) ) ) ) ) ) ) )
= ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ ( inj__ty_2Erealax_2Ereal @ V8c ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) @ ( inj__ty_2Erealax_2Ereal @ V9d ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__INTERVAL__OPEN__INTERVAL,axiom,
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ? [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ V0s ) ) ) ) ) )
& ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V4s ) )
<=> ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V4s ) )
=> ? [V6a: tp__ty_2Erealax_2Ereal,V7b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) ) ) @ V4s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__INTERVAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ? [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__OPEN__INTERVAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ? [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIAMETER__INTERVAL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ediameter @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ediameter @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIMAGE__TWIZZLE__INTERVAL,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ A_27a )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : V3x ) )
@ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEQ__INTERVAL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
<=> ( ( ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
| ( ( V0a = V2c )
& ( V1b = V3d ) ) ) )
& ! [V4a: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal,V6c: tp__ty_2Erealax_2Ereal,V7d: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6c ) ) @ ( inj__ty_2Erealax_2Ereal @ V7d ) ) ) )
<=> ( ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6c ) ) @ ( inj__ty_2Erealax_2Ereal @ V7d ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
& ! [V8a: tp__ty_2Erealax_2Ereal,V9b: tp__ty_2Erealax_2Ereal,V10c: tp__ty_2Erealax_2Ereal,V11d: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V10c ) ) @ ( inj__ty_2Erealax_2Ereal @ V11d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
<=> ( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V10c ) ) @ ( inj__ty_2Erealax_2Ereal @ V11d ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
& ! [V12a: tp__ty_2Erealax_2Ereal,V13b: tp__ty_2Erealax_2Ereal,V14c: tp__ty_2Erealax_2Ereal,V15d: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12a ) ) @ ( inj__ty_2Erealax_2Ereal @ V13b ) ) )
= ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V14c ) ) @ ( inj__ty_2Erealax_2Ereal @ V15d ) ) ) )
<=> ( ( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12a ) ) @ ( inj__ty_2Erealax_2Ereal @ V13b ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V14c ) ) @ ( inj__ty_2Erealax_2Ereal @ V15d ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
| ( ( V12a = V14c )
& ( V13b = V15d ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL__EQ,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) )
<=> ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__EQ,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPACT__INTERVAL__EQ,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) )
<=> ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEQ__BALLS,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1a_27: tp__ty_2Erealax_2Ereal,V2r: tp__ty_2Erealax_2Ereal,V3r_27: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) )
= ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3r_27 ) ) ) )
<=> ( ( ( V0a = V1a_27 )
& ( V2r = V3r_27 ) )
| ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3r_27 ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) )
& ! [V4a: tp__ty_2Erealax_2Ereal,V5a_27: tp__ty_2Erealax_2Ereal,V6r: tp__ty_2Erealax_2Ereal,V7r_27: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6r ) ) )
= ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5a_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V7r_27 ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V7r_27 ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
& ! [V8a: tp__ty_2Erealax_2Ereal,V9a_27: tp__ty_2Erealax_2Ereal,V10r: tp__ty_2Erealax_2Ereal,V11r_27: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V10r ) ) )
= ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9a_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V11r_27 ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V10r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V11r_27 ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
& ! [V12a: tp__ty_2Erealax_2Ereal,V13a_27: tp__ty_2Erealax_2Ereal,V14r: tp__ty_2Erealax_2Ereal,V15r_27: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12a ) ) @ ( inj__ty_2Erealax_2Ereal @ V14r ) ) )
= ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V13a_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V15r_27 ) ) ) )
<=> ( ( ( V12a = V13a_27 )
& ( V14r = V15r_27 ) )
| ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V14r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V15r_27 ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERVAL__CASES,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
| ( V2x = V0a )
| ( V2x = V1b ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__CLOSED__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__OPEN__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
=> ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
& ( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) )
= ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESPHERE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__SPHERE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__INTERVAL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__INTERVAL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
= ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__INTERVAL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1e: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBALL__INTERVAL__0,axiom,
! [V0e: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) )
= ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECBALL__INTERVAL__0,axiom,
! [V0e: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) )
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__DIFF__OPEN__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERVAL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Emidpoint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEGMENT,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__SEGMENT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEGMENT__SCALAR__MULTIPLE,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2v: tp__ty_2Erealax_2Ereal] :
( ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2v ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2v ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ V3x ) @ ( inj__ty_2Erealax_2Ereal @ V2v ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V3x ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ V3x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V3x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) )
& ! [V4a: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal,V6v: tp__ty_2Erealax_2Ereal] :
( ( V6v
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6v ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) @ ( inj__ty_2Erealax_2Ereal @ V6v ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ V7x ) @ ( inj__ty_2Erealax_2Ereal @ V6v ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ V7x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V7x ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) @ V7x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V7x ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eis__interval,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V0s ) )
<=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ V0s ) ) )
=> ( ( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
| ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__EMPTY,axiom,
p @ ( ap @ c_2Ereal__topology_2Eis__interval @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__UNIV,axiom,
p @ ( ap @ c_2Ereal__topology_2Eis__interval @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__POINTWISE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V0s ) )
=> ( ? [V2a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ V0s ) )
& ( V2a = V1x ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) ) )
<=> ? [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V0s ) )
<=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__CASES,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V0s ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( V0s
= ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) )
| ? [V1a: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V2x ) ) ) ) )
| ? [V3a: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ V4x ) ) ) ) )
| ? [V5b: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V6x ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) ) ) )
| ? [V7b: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V8x ) @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) ) ) ) )
| ? [V9a: tp__ty_2Erealax_2Ereal,V10b: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V11x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V11x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V9a ) ) @ V11x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V11x ) @ ( inj__ty_2Erealax_2Ereal @ V10b ) ) ) ) ) ) )
| ? [V12a: tp__ty_2Erealax_2Ereal,V13b: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V14x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V14x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V12a ) ) @ V14x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V14x ) @ ( inj__ty_2Erealax_2Ereal @ V13b ) ) ) ) ) ) )
| ? [V15a: tp__ty_2Erealax_2Ereal,V16b: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V17x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V17x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V15a ) ) @ V17x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V17x ) @ ( inj__ty_2Erealax_2Ereal @ V16b ) ) ) ) ) ) )
| ? [V18a: tp__ty_2Erealax_2Ereal,V19b: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V20x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V20x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V18a ) ) @ V20x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V20x ) @ ( inj__ty_2Erealax_2Ereal @ V19b ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V1t ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERVAL__SUBSET__IS__INTERVAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ V0s ) )
<=> ( ( ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERVAL__CONTAINS__COMPACT__NEIGHBOURHOOD,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) )
=> ? [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) ) @ V0s ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__SCALING,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V0s ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Eis__interval
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ V2x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__SCALING__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap @ c_2Ereal__topology_2Eis__interval
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ V2x ) ) )
@ V0s ) ) )
<=> ( ( V1c
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECARD__FRONTIER__INTERVAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V0s ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__COMPONENT__UBOUND,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4k: $i] :
( ( mem @ V4k @ A_27b )
=> ( ( ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
@ V0net ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__COMPONENT__LBOUND,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( ap @ V1f @ V4x ) ) ) )
@ V0net ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__COMPONENT__EQ,axiom,
! [A_27a: del,A_27b: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2i: $i] :
( ( mem @ V2i @ A_27b )
=> ! [V3l: tp__ty_2Erealax_2Ereal,V4b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) )
& ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) )
@ V0net ) ) )
=> ( V3l = V4b ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__COMPONENT__LE,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal,V4m: tp__ty_2Erealax_2Ereal] :
( ( ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V5x ) ) @ ( ap @ V2g @ V5x ) ) ) )
@ V0net ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__DROP__LE,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal,V4m: tp__ty_2Erealax_2Ereal] :
( ( ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V0net ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V5x ) ) @ ( ap @ V2g @ V5x ) ) ) )
@ V0net ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__DROP__UBOUND,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) )
& ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
@ V0net ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__DROP__LBOUND,axiom,
! [A_27a: del,V0net: $i] :
( ( mem @ V0net @ ( ty_2Ereal__topology_2Enet @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ A_27a ) @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0net ) )
& ~ ( p @ ( ap @ ( c_2Ereal__topology_2Etrivial__limit @ A_27a ) @ V0net ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( ap @ V1f @ V4x ) ) ) )
@ V0net ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIMAGE__CLOSURE__SUBSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__IMAGE__CLOSURE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
= ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__IMAGE__BOUNDED,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__ABS__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__COMPONENT__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__COMPONENT__GE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__MAP__CLOSURES,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) )
<=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__UNION,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V3s ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Erealax_2Ereal ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Ewithin @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ V4t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V3s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V4t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V3s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V2g ) @ V4t ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V3s ) )
& ~ ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) )
| ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V4t ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ V0P @ V6x ) ) @ ( ap @ V1f @ V6x ) ) @ ( ap @ V2g @ V6x ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__LOCAL,axiom,
! [V0t: $i] :
( ( mem @ V0t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V2s ) @ V0t ) ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V2s ) @ V0t ) ) ) @ V0t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V0t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V2s ) @ V0t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LOCAL,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) ) ) @ V3s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) ) ) @ V4t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V3s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V2g ) @ V4t ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V3s ) )
& ~ ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) )
| ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V4t ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ V0P @ V6x ) ) @ ( ap @ V1f @ V6x ) ) @ ( ap @ V2g @ V6x ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4a: tp__ty_2Erealax_2Ereal] :
( ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5t: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5t ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5t ) @ V3s ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V2h @ V5t ) ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) ) ) ) ) ) )
& ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6t: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6t ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6t ) @ V3s ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( ap @ V2h @ V6t ) ) ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V2h ) @ V3s ) )
& ! [V7t: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7t ) ) @ V3s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2h @ ( inj__ty_2Erealax_2Ereal @ V7t ) ) )
= V4a ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V7t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V7t ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8t: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V2h @ V8t ) ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) ) @ ( ap @ V0f @ V8t ) ) @ ( ap @ V1g @ V8t ) ) ) )
@ V3s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__1,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal] :
( ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4t: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4t ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4t ) @ V2s ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V4t ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) ) ) ) ) )
& ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5t: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5t ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5t ) @ V2s ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ V5t ) ) ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6t: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V6t ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) @ ( ap @ V0f @ V6t ) ) @ ( ap @ V1g @ V6t ) ) ) )
@ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEXTENSION__FROM__CLOPEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V2t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2t ) ) @ V3u ) )
& ( ( V3u
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
=> ( V1s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
=> ? [V4g: $i] :
( ( mem @ V4g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V4g ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V4g ) @ V1s ) ) @ V3u ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V2t ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V4g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__IVT__HYPERPLANE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal,V3a: tp__ty_2Erealax_2Ereal,V4b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) )
=> ? [V5z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ V0s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) )
= V4b ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__IVT__COMPONENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal,V3a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
=> ? [V4z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4z ) ) @ V0s ) )
& ( V4z = V3a ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EMAPPING__CONNECTED__ONTO__SEGMENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
& ~ ? [V3a: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
=> ? [V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V4f ) @ V0s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V4f ) @ V0s )
= ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__INCREASING__CONVERGENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ V0s @ V1n ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V1n ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ V0s @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) )
=> ? [V3l: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__DECREASING__CONVERGENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ V0s @ V1n ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V1n ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0s @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
=> ? [V3l: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ehomeomorphism,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V2f ) @ V3g ) ) )
<=> ( ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V3g @ ( ap @ V2f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
= V4x ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2f ) @ V0s )
= V1t )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V2f ) @ V0s ) )
& ! [V5y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V1t ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2f @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) )
= V5y ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V3g ) @ V1t )
= V0s )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V3g ) @ V1t ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ehomeomorphic,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V1t ) )
<=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V2f ) @ V3g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V2f ) @ V3g ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V2f ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2f ) @ V0s ) ) @ V1t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V3g ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V3g ) @ V1t ) ) @ V0s ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V3g @ ( ap @ V2f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
= V4x ) )
& ! [V5y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V1t ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2f @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) )
= V5y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__OF__SUBSETS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4s_27: $i] :
( ( mem @ V4s_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V5t_27: $i] :
( ( mem @ V5t_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V1g ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4s_27 ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5t_27 ) @ V3t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V4s_27 )
= V5t_27 ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V4s_27 ) @ V5t_27 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V1g ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__ID,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V0s ) )
@ ( ap
@ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : V1x ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : V2x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__REFL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__SYM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V1g ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3t ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V1g ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SYM,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V1t ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__COMPOSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V5t: $i] :
( ( mem @ V5t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V6u: $i] :
( ( mem @ V6u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V4s ) @ V5t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V1g ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V5t ) @ V6u ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V2h ) @ V3k ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V4s ) @ V6u ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2h ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3k ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__IMP__CARD__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__FINITENESS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V1t ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__EMPTY,axiom,
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) @ V1s ) )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__MINIMAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V1t ) )
<=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V2f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ V1t ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V3g @ ( ap @ V2f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
= V4x ) ) )
& ! [V5y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V1t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) @ V0s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2f @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) )
= V5y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V2f ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V3g ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__SELF,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) )
=> ( V2x = V3y ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__LEFT__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) )
=> ( V3x = V4y ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__RIGHT__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) )
=> ( V3x = V4y ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2t ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__SELF,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Ehomeomorphic
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) )
@ V1s ) )
@ V1s ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__LEFT__EQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Ehomeomorphic
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) )
@ V1s ) )
@ V2t ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V1s ) @ V2t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__RIGHT__EQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V1s )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) )
@ V2t ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V1s ) @ V2t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__QUOTIENT__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V1g ) ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4u ) @ V3t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ V4u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ V4u ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING__LEFT,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Ehomeomorphic
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ V3x ) ) )
@ V1s ) )
@ V2t ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V1s ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING__RIGHT,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V1s )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ V3x ) ) )
@ V2t ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V1s ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1t ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V1t ) )
<=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V0s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__FINITE__STRONG,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
| ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1t ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V1t ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1t ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V0s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal ) @ V1t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIFT__TO__QUOTIENT__SPACE__UNIQUE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s )
= V3t )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V2s )
= V4u )
& ! [V5v: $i] :
( ( mem @ V5v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5v ) @ V3t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V6x ) ) @ V5v ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ V5v ) ) ) ) )
& ! [V7v: $i] :
( ( mem @ V7v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V7v ) @ V4u ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V1g @ V8x ) ) @ V7v ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V4u ) ) @ V7v ) ) ) ) )
& ! [V9x: tp__ty_2Erealax_2Ereal,V10y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V10y ) ) @ V2s ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V10y ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V10y ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V3t ) @ V4u ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE__OPEN__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s )
= V3t )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
= V4x ) )
& ! [V5u: $i] :
( ( mem @ V5u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V5u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5u ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE__CLOSED__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s )
= V3t )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
= V4x ) )
& ! [V5u: $i] :
( ( mem @ V5u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V5u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5u ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__OPEN__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) )
=> ( V3x = V4y ) )
& ! [V5u: $i] :
( ( mem @ V5u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V5u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5u ) ) ) ) ) )
=> ? [V6g: $i] :
( ( mem @ V6g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V6g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__CLOSED__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) )
=> ( V3x = V4y ) )
& ! [V5u: $i] :
( ( mem @ V5u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V5u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5u ) ) ) ) ) )
=> ? [V6g: $i] :
( ( mem @ V6g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V6g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__OPEN__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V1g ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V4u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V4u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__CLOSED__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V1g ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) ) @ V4u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V4u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__OPEN__MAP__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) )
=> ( V3x = V4y ) ) )
=> ( ? [V5g: $i] :
( ( mem @ V5g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V5g ) ) ) )
<=> ! [V6u: $i] :
( ( mem @ V6u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V6u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V6u ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__CLOSED__MAP__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) )
=> ( V3x = V4y ) ) )
=> ( ? [V5g: $i] :
( ( mem @ V5g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V5g ) ) ) )
<=> ! [V6u: $i] :
( ( mem @ V6u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V6u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V6u ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINJECTIVE__MAP__OPEN__IFF__CLOSED,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) )
=> ( V3x = V4y ) ) )
=> ( ! [V5u: $i] :
( ( mem @ V5u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V5u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5u ) ) ) ) )
<=> ! [V6u: $i] :
( ( mem @ V6u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V6u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V6u ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__IMP__CLOSED__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__IMP__QUOTIENT__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3u ) @ V2t ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ V3u ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V3u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V2s ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
= V3x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V0s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1f ) @ V0s )
= V2t )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V0s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) )
=> ( V3x = V4y ) ) )
=> ? [V5g: $i] :
( ( mem @ V5g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V1f ) @ V5g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V0s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1f ) @ V0s )
= V2t )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V0s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) )
=> ( V3x = V4y ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__FROM__COMPOSITION__SURJECTIVE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s )
= V3t )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3t ) ) @ V4u ) )
& ? [V5h: $i] :
( ( mem @ V5h @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V4u ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) ) @ V5h ) ) ) ) )
=> ( ? [V6f_27: $i] :
( ( mem @ V6f_27 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V6f_27 ) ) ) )
& ? [V7g_27: $i] :
( ( mem @ V7g_27 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3t ) @ V4u ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V1g ) @ V7g_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__FROM__COMPOSITION__INJECTIVE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V3t ) ) @ V4u ) )
& ! [V5x: tp__ty_2Erealax_2Ereal,V6y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) @ V3t ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) ) ) )
=> ( V5x = V6y ) )
& ? [V7h: $i] :
( ( mem @ V7h @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V4u ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1g ) @ V0f ) ) @ V7h ) ) ) ) )
=> ( ? [V8f_27: $i] :
( ( mem @ V8f_27 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V8f_27 ) ) ) )
& ? [V9g_27: $i] :
( ( mem @ V9g_27 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3t ) @ V4u ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V1g ) @ V9g_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__COMPACTNESS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V1t ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__CONNECTEDNESS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V1t ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( V1c
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ V2x ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V2x ) ) )
@ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__AFFINITY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( V2c
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ V3x ) ) ) )
@ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__BALLS__CBALL__SPHERE,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: tp__ty_2Erealax_2Ereal,V3e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) )
& ! [V4a: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal,V6d: tp__ty_2Erealax_2Ereal,V7e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6d ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7e ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6d ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) @ ( inj__ty_2Erealax_2Ereal @ V7e ) ) ) ) ) )
& ! [V8a: tp__ty_2Erealax_2Ereal,V9b: tp__ty_2Erealax_2Ereal,V10d: tp__ty_2Erealax_2Ereal,V11e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V10d ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V11e ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V10d ) ) ) ) @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) @ ( inj__ty_2Erealax_2Ereal @ V11e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__BALLS,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: tp__ty_2Erealax_2Ereal,V3e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__CBALL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: tp__ty_2Erealax_2Ereal,V3e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SPHERE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: tp__ty_2Erealax_2Ereal,V3e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) ) @ ( ap @ c_2Ereal__topology_2Esphere @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__ONE__POINT__COMPACTIFICATIONS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ V1t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Erealax_2Ereal ) @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Erealax_2Ereal ) @ V1t ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__OPEN__INTERVALS,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__OPEN__INTERVAL__UNIV,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__REAL,axiom,
p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ty_2Erealax_2Ereal @ ( arr @ ty_2Enum_2Enum @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2EUNIV @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNCOUNTABLE__REAL,axiom,
~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__REAL__IMP__UNCOUNTABLE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Erealax_2Ereal ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__EUCLIDEAN,axiom,
p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2EUNCOUNTABLE__EUCLIDEAN,axiom,
~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__INTERVAL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNCOUNTABLE__INTERVAL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOUNTABLE__OPEN__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
<=> ( ( ap @ c_2Ereal__topology_2EOPEN__interval @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__OPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNCOUNTABLE__OPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Erealax_2Ereal ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__BALL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__CBALL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__IMP__NOT__OPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ~ ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IMP__INFINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EEMPTY__INTERIOR__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ V0s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__CBALL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__BALL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1r: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__DISCRETE__FINITE__RANGE__CONSTANT__EQ,axiom,
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1f ) @ V0s ) ) @ V2t ) )
& ! [V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V2t ) )
=> ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V2t ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
=> ? [V4a: tp__ty_2Erealax_2Ereal] :
! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= V4a ) ) ) ) ) ) )
& ! [V6s: $i] :
( ( mem @ V6s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V6s ) )
<=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V7f ) @ V6s ) )
& ! [V8x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ V6s ) )
=> ? [V9e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) )
& ! [V10y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V10y ) ) @ V6s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V7f @ ( inj__ty_2Erealax_2Ereal @ V10y ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ V7f @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V7f @ ( inj__ty_2Erealax_2Ereal @ V10y ) ) ) @ ( ap @ V7f @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) ) ) ) ) ) ) )
=> ? [V11a: tp__ty_2Erealax_2Ereal] :
! [V12x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12x ) ) @ V6s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V7f @ ( inj__ty_2Erealax_2Ereal @ V12x ) ) )
= V11a ) ) ) ) ) )
& ! [V13s: $i] :
( ( mem @ V13s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V13s ) )
<=> ! [V14f: $i] :
( ( mem @ V14f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V14f ) @ V13s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V14f ) @ V13s ) ) ) )
=> ? [V15a: tp__ty_2Erealax_2Ereal] :
! [V16x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V16x ) ) @ V13s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V14f @ ( inj__ty_2Erealax_2Ereal @ V16x ) ) )
= V15a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__RANGE__CONSTANT__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1f ) @ V0s ) ) @ V2t ) )
& ! [V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V2t ) )
=> ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V2t ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
=> ? [V4a: tp__ty_2Erealax_2Ereal] :
! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= V4a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCRETE__RANGE__CONSTANT__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V0s ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ? [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V0s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ) )
=> ? [V5a: tp__ty_2Erealax_2Ereal] :
! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
= V5a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__FINITE__RANGE__CONSTANT__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V0s ) )
<=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1f ) @ V0s ) ) ) )
=> ? [V2a: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= V2a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__RANGE__CONSTANT,axiom,
! [V0t: $i] :
( ( mem @ V0t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1f ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1f ) @ V2s ) ) @ V0t ) )
& ! [V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V0t ) )
=> ( ( ap @ ( ap @ c_2Ereal__topology_2Econnected__component @ V0t ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
=> ? [V4a: tp__ty_2Erealax_2Ereal] :
! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= V4a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCRETE__RANGE__CONSTANT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
=> ? [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ) )
=> ? [V5a: tp__ty_2Erealax_2Ereal] :
! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
= V5a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__FINITE__RANGE__CONSTANT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) )
=> ? [V2a: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= V2a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__HYPERPLANES,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( ( V0a
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V2c
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Ehomeomorphic
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__HYPERPLANE__STANDARD__HYPERPLANE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( V0a
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Ehomeomorphic
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V4x ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__STANDARD__HYPERPLANE__HYPERPLANE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( V0a
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Ehomeomorphic
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V3x ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECAUCHY__ISOMETRIC,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2e: tp__ty_2Erealax_2Ereal,V3x: $i] :
( ( mem @ V3x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) )
& ! [V5n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V3x @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3x ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecauchy @ V3x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPLETE__ISOMETRIC__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecomplete @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecomplete @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINJECTIVE__IMP__ISOMETRIC,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( V2x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
=> ? [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__INJECTIVE__IMAGE__SUBSPACE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Esubspace @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( V2x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__SURJECTIVE__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V1y: tp__ty_2Erealax_2Ereal] :
? [V2x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= V1y ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V3s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__BIJECTIVE__LINEAR__IMAGE__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) )
=> ( V2x = V3y ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
? [V5x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= V4y ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
=> ( V1x = V2y ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V3s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__INJECTIVE__LINEAR__IMAGE__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) )
=> ( V2x = V3y ) ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__LINEAR__IMAGE__SUBSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) )
=> ( V2x = V3y ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__BOUNDED__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1s ) ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__INTERIOR__IMAGE__SUBSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) )
=> ( V2x = V3y ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__IMAGE__SUBSET__INTERIOR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2y: tp__ty_2Erealax_2Ereal] :
? [V3x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= V2y ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1s ) ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__BIJECTIVE__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) )
=> ( V2x = V3y ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
? [V5x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= V4y ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIN__INTERIOR__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1g ) )
& ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1g )
= ( c_2Ecombin_2EI @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V2s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELINEAR__OPEN__MAPPING,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1g ) )
& ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1g )
= ( c_2Ecombin_2EI @ ty_2Erealax_2Ereal ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) )
=> ( V2x = V3y ) ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ ( ap @ c_2Ereal__topology_2Einterior @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPLETE__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
=> ( V1x = V2y ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecomplete @ V3s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ecomplete @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECOMPLETE__INJECTIVE__LINEAR__IMAGE__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) )
=> ( V2x = V3y ) ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecomplete @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Ecomplete @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELIMPT__INJECTIVE__LINEAR__IMAGE__EQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1f ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) )
=> ( V3x = V4y ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( ap @ V1f @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V1f ) @ V2s ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elimit__point__of @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINTERIOR__NEGATIONS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V1x ) ) )
@ V0s ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V2x ) ) )
@ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESYMMETRIC__INTERIOR,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V0s ) ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSURE__NEGATIONS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ap @ c_2Ereal__topology_2Eclosure
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V1x ) ) )
@ V0s ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ V2x ) ) )
@ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESYMMETRIC__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ V0s ) ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__SUBSTANDARD,axiom,
( p
@ ( ap @ c_2Ereal__topology_2Esubspace
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V0x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V0x ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__SUBSTANDARD,axiom,
( p
@ ( ap @ c_2Ereal__topology_2EClosed
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V0x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V0x ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIM__SUBSTANDARD,axiom,
( ( surj__ty_2Enum_2Enum
@ ( ap @ c_2Ereal__topology_2Edim
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V0x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V0x ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) )
= fo__c_2Enum_2E0 ) ).
thf(conj_thm_2Ereal__topology_2EAFFINITY__INVERSES,axiom,
! [V0m: tp__ty_2Erealax_2Ereal,V1c: tp__ty_2Erealax_2Ereal] :
( ( V0m
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ( ap
@ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) @ V3x ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) ) ) )
= ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : V4x ) )
& ( ( ap
@ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) @ V5x ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) )
= ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : V7x ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__AFFINITY__LE,axiom,
! [V0m: tp__ty_2Erealax_2Ereal,V1c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LE__AFFINITY,axiom,
! [V0m: tp__ty_2Erealax_2Ereal,V1c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__AFFINITY__LT,axiom,
! [V0m: tp__ty_2Erealax_2Ereal,V1c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LT__AFFINITY,axiom,
! [V0m: tp__ty_2Erealax_2Ereal,V1c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__AFFINITY__EQ,axiom,
! [V0m: tp__ty_2Erealax_2Ereal,V1c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( V0m
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) )
= V3y )
<=> ( V2x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__EQ__AFFINITY,axiom,
! [V0m: tp__ty_2Erealax_2Ereal,V1c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( V0m
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( V3y
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) ) ) )
= V2x ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIMAGE__AFFINITY__INTERVAL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2m: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2m ) ) @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) )
@ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2m ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2m ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2m ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2ECLOSED__interval @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2m ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2m ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Esums,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ V2s ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V2s ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V3n ) ) ) @ V0f ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1l ) )
@ c_2Ereal__topology_2Esequentially ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Einfsum,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V0s ) @ V1f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2l: $i] : ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V1f ) @ V2l ) @ V0s ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Esummable,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V0s ) @ V1f ) )
<=> ? [V2l: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__SUMMABLE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__INFSUM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V1s ) @ V0f ) ) @ V1s ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__LIM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f )
@ ( ap @ ( ap @ ( c_2Ereal__topology_2Elim__def @ ty_2Enum_2Enum ) @ c_2Ereal__topology_2Esequentially )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1s ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V2n ) ) ) @ V0f ) ) ) )
@ V1s ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFINITE__INTER__NUMSEG,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V0s ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__FROM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V3n ) ) @ V0f ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1l ) )
@ c_2Ereal__topology_2Esequentially ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__UNIQUE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2l_27: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ V3s ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l_27 ) ) @ V3s ) ) )
=> ( V1l = V2l_27 ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINFSUM__UNIQUE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V2s ) @ V0f ) )
= V1l ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__TERMS__TOZERO,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ c_2Ereal__topology_2Esequentially ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__FINITE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ V1s ) @ V0f ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__LINEAR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V3s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1h ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Ereal__topology_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ V1h @ ( ap @ V0f @ V4n ) ) ) )
@ ( ap @ V1h @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) )
@ V3s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__0,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Ereal__topology_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ V0s ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__ADD,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal,V2y: $i] :
( ( mem @ V2y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3y0: tp__ty_2Erealax_2Ereal,V4s: $i] :
( ( mem @ V4s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ V4s ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V2y ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) @ V4s ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Ereal__topology_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0x @ V5n ) ) @ ( ap @ V2y @ V5n ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) )
@ V4s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__SUB,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal,V2y: $i] :
( ( mem @ V2y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3y0: tp__ty_2Erealax_2Ereal,V4s: $i] :
( ( mem @ V4s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ V4s ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V2y ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) @ V4s ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Ereal__topology_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0x @ V5n ) ) @ ( ap @ V2y @ V5n ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) )
@ V4s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__CMUL,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ V3s ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Ereal__topology_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ V0x @ V4n ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
@ V3s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__NEG,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ V2s ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Ereal__topology_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0x @ V3n ) ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
@ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__IFF,axiom,
! [V0l: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ! [V4x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4x ) ) @ V3k ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Enum_2Enum @ V4x ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ V3k ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ V3k ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__EQ,axiom,
! [V0l: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ! [V4x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4x ) ) @ V3k ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2g @ ( inj__ty_2Enum_2Enum @ V4x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V1f ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ V3k ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V2g ) @ ( inj__ty_2Erealax_2Ereal @ V0l ) ) @ V3k ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__0,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__FINITE__SUPPORT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1s ) @ V2k ) ) )
& ! [V3x: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ V2k ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ V1s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1s ) @ V2k ) ) @ V0f ) ) @ V2k ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__COMPONENT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V1s ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Ereal__topology_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ V0f @ V3i ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2l ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__DIFFS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ c_2Ereal__topology_2Esequentially ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Ereal__topology_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V2n ) ) @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V2n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
@ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1k ) ) )
@ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__TRIVIAL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__RESTRICT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Ereal__topology_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V3n ) @ V1k ) ) @ ( ap @ V0f @ V3n ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2l ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V1k ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__SUM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2k: $i] :
( ( mem @ V2k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V3s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Enum_2Enum ) @ V3s ) @ V2k ) )
& ! [V4x: tp__ty_2Enum_2Enum] :
( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4x ) ) @ V3s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ V3s ) @ V0f ) )
= V1l ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ V2k ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__REINDEX,axiom,
! [V0k: tp__ty_2Enum_2Enum,V1a: $i] :
( ( mem @ V1a @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3n: tp__ty_2Enum_2Enum] :
( ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Ereal__topology_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4x: $i] : ( ap @ V1a @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V4x ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2l ) )
@ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V1a ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( ap @ c_2Ereal__topology_2Efrom @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__REINDEX__GEN,axiom,
! [V0k: tp__ty_2Enum_2Enum,V1a: $i] :
( ( mem @ V1a @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Ereal__topology_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4x: $i] : ( ap @ V1a @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V4x ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2l ) )
@ V3s ) )
<=> ( p
@ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V1a ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5i: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V5i ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) )
@ V3s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__LINEAR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1h ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ V1h @ ( ap @ V0f @ V3n ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__0,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V0s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__ADD,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V0x ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V1y ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0x @ V3n ) ) @ ( ap @ V1y @ V3n ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__SUB,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V0x ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V1y ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0x @ V3n ) ) @ ( ap @ V1y @ V3n ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__CMUL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V0s ) @ V1x ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V0s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ V1x @ V3n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__NEG,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s ) @ V0x ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0x @ V2n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__IFF,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ! [V3x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ V2k ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2k ) @ V0f ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2k ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ! [V3x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ V2k ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2k ) @ V0f ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2k ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__COMPONENT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s ) @ V0f ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2i: $i] : ( ap @ V0f @ V2i ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__SUBSET,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Enum_2Enum ) @ V1s ) @ V2t ) )
& ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Ereal__topology_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V4i ) @ V1s ) ) @ ( ap @ V0x @ V4i ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V3l ) )
@ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__SUBSET,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Enum_2Enum ) @ V1s ) @ V2t ) )
& ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2t )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V3i ) @ V1s ) ) @ ( ap @ V0x @ V3i ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s ) @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__TRIVIAL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) @ V0f ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__RESTRICT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V2n ) @ V1k ) ) @ ( ap @ V0f @ V2n ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1k ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__FINITE__DIFF,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Enum_2Enum ) @ V1t ) @ V2s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ V1t ) @ V0f ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Enum_2Enum ) @ V2s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__FINITE__UNION,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Enum_2Enum ) @ V2t ) @ V1s ) ) @ V0f ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__OFFSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V0f ) ) ) @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMS__OFFSET__REV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V0f ) ) ) @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__REINDEX,axiom,
! [V0k: tp__ty_2Enum_2Enum,V1a: $i] :
( ( mem @ V1a @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3x: $i] : ( ap @ V1a @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V3x ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( ap @ c_2Ereal__topology_2Efrom @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) ) @ V1a ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__DROP__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal,V4b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) @ V2s ) )
& ! [V5x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V5x ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V5x ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V5x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__DROP__POS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ V1s ) )
& ! [V3x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__BOUND,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal,V4b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) @ V2s ) )
& ! [V5i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V5i ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V5i ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V5i ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINFSUM__LINEAR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V1h ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V2s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ V1h @ ( ap @ V0f @ V3n ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1h @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V2s ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINFSUM__0,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V0s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1i: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINFSUM__ADD,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V0x ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V1y ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V2s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0x @ V3i ) ) @ ( ap @ V1y @ V3i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V2s ) @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V2s ) @ V1y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINFSUM__SUB,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V0x ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V1y ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V2s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0x @ V3i ) ) @ ( ap @ V1y @ V3i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V2s ) @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V2s ) @ V1y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINFSUM__CMUL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V0s ) @ V1x ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V0s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ V1x @ V3n ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V0s ) @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINFSUM__NEG,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V0s ) @ V1x ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V0s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V1x @ V2n ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V0s ) @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINFSUM__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2k ) @ V0f ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2k ) @ V1g ) )
& ! [V3x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ V2k ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V3x ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V2k ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V2k ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EINFSUM__RESTRICT,axiom,
! [V0k: $i] :
( ( mem @ V0k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V2n ) @ V0k ) ) @ ( ap @ V1a @ V2n ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V0k ) @ V1a ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPARTIAL__SUMS__COMPONENT__LE__INFSUM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ! [V3i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3i ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s ) @ V0f ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1s ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ V0f ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V1s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EPARTIAL__SUMS__DROP__LE__INFSUM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ! [V3i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3i ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s ) @ V0f ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1s ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ V0f ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Einfsum @ V1s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEQUENCE__CAUCHY__WLOG,axiom,
! [V0e: tp__ty_2Erealax_2Ereal,V1P: $i] :
( ( mem @ V1P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ V1P @ ( inj__ty_2Enum_2Enum @ V3m ) ) )
& ( p @ ( ap @ V1P @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V2s @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V2s @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) ) )
<=> ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ V1P @ ( inj__ty_2Enum_2Enum @ V5m ) ) )
& ( p @ ( ap @ V1P @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V2s @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) @ ( ap @ V2s @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUM__DIFF__LEMMA,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1k ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1k ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) @ V0f ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1k ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EABS__SUM__TRIVIAL__LEMMA,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum,V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) )
=> ( ( ( p @ V0P )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1s ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V4f ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) ) )
<=> ( ( p @ V0P )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1s ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V4f ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__CAUCHY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ? [V2l: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ V1s ) )
<=> ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4N: tp__ty_2Enum_2Enum] :
! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V4N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1s ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ V0f ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__CAUCHY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s ) @ V0f ) )
<=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ? [V3N: tp__ty_2Enum_2Enum] :
! [V4m: tp__ty_2Enum_2Enum,V5n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V3N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1s ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) @ V0f ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__IFF__EVENTUALLY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ? [V3N: tp__ty_2Enum_2Enum] :
! [V4n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3N ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ V2k ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2k ) @ V0f ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2k ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__EQ__EVENTUALLY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ? [V3N: tp__ty_2Enum_2Enum] :
! [V4n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3N ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ V2k ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2k ) @ V0f ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2k ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__IFF__COFINITE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Enum_2Enum ) @ V1s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Enum_2Enum ) @ V2t ) @ V1s ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s ) @ V0f ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2t ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__EQ__COFINITE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Enum_2Enum ) @ V1s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Enum_2Enum ) @ V2t ) @ V1s ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s ) @ V0f ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2t ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__FROM__ELSEWHERE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ V0f ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__CAUCHY__UNIFORM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ? [V3l: $i] :
( ( mem @ V3l @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
& ! [V4e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
=> ? [V5N: tp__ty_2Enum_2Enum] :
! [V6n: tp__ty_2Enum_2Enum,V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V5N ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
& ( p @ ( ap @ V0P @ V7x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V2k ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ ( ap @ V1f @ V7x ) ) ) @ ( ap @ V3l @ V7x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) ) ) ) )
<=> ! [V8e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V8e ) ) )
=> ? [V9N: tp__ty_2Enum_2Enum] :
! [V10m: tp__ty_2Enum_2Enum,V11n: tp__ty_2Enum_2Enum,V12x: $i] :
( ( mem @ V12x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V9N ) ) @ ( inj__ty_2Enum_2Enum @ V10m ) ) )
& ( p @ ( ap @ V0P @ V12x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V2k ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V10m ) ) @ ( inj__ty_2Enum_2Enum @ V11n ) ) ) ) @ ( ap @ V1f @ V12x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V8e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__GOESTOZERO,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V0s ) @ V1x ) )
=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V3n ) @ V0s ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V1x @ V3n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) )
@ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__IMP__TOZERO,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1k ) @ V0f ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V2n ) @ V1k ) ) @ ( ap @ V0f @ V2n ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ c_2Ereal__topology_2Esequentially ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__IMP__BOUNDED,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1k ) @ V0f ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ V0f ) @ V1k ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__IMP__SUMS__BOUNDED,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) @ V0f ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2n ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V2n ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__COMPARISON,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ? [V3l: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V2s ) )
& ? [V4N: tp__ty_2Enum_2Enum] :
! [V5n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ ( inj__ty_2Enum_2Enum @ V4N ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) ) )
=> ? [V6l: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V6l ) ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__COMPARISON,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V1g ) )
& ? [V3N: tp__ty_2Enum_2Enum] :
! [V4n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( inj__ty_2Enum_2Enum @ V3N ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__ABSCONV__IMP__CONV,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1k )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0x @ V2n ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1k ) @ V0x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__SUBSET__ABSCONV,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0x @ V3n ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Enum_2Enum ) @ V2t ) @ V1s ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V2t )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0x @ V4n ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__COMPARISON__BOUND,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ V2s ) )
& ! [V4i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) ) )
=> ? [V5l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V5l ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V5l ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__COMPARISON__UNIFORM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ? [V4l: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V4l ) ) @ V3s ) )
& ? [V5N: tp__ty_2Enum_2Enum] :
! [V6n: tp__ty_2Enum_2Enum,V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V5N ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ V3s ) )
& ( p @ ( ap @ V2P @ V7x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ V0f @ V7x ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ) )
=> ? [V8l: $i] :
( ( mem @ V8l @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
& ! [V9e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) )
=> ? [V10N: tp__ty_2Enum_2Enum] :
! [V11n: tp__ty_2Enum_2Enum,V12x: $i] :
( ( mem @ V12x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V10N ) ) @ ( inj__ty_2Enum_2Enum @ V11n ) ) )
& ( p @ ( ap @ V2P @ V12x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V3s ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V11n ) ) ) ) @ ( ap @ V0f @ V12x ) ) ) @ ( ap @ V8l @ V12x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__RATIO,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1a: $i] :
( ( mem @ V1a @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3N: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( inj__ty_2Enum_2Enum @ V3N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V1a @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V1a @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ) )
=> ? [V5l: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V1a ) @ ( inj__ty_2Erealax_2Ereal @ V5l ) ) @ V2s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBOUNDED__PARTIAL__SUMS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2n ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V2n ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3m: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ V3m ) @ V4n ) ) @ V0f ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V3m ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V4n ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__BILINEAR__PARTIAL__PRE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal,V4k: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V2h ) )
& ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ V2h @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V5n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V1g @ V5n ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V3l ) )
@ c_2Ereal__topology_2Esequentially ) )
& ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V4k ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6n: $i] : ( ap @ ( ap @ V2h @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V6n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V0f @ V6n ) ) ) @ ( ap @ V1g @ V6n ) ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V4k ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7n: $i] : ( ap @ ( ap @ V2h @ ( ap @ V0f @ V7n ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1g @ V7n ) ) @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V7n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__DIRICHLET__BILINEAR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V3k: tp__ty_2Enum_2Enum,V4m: tp__ty_2Enum_2Enum,V5p: tp__ty_2Enum_2Enum,V6l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ebilinear @ V2h ) )
& ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ V7n ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V7n ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) )
& ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V5p ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V8n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V1g @ V8n ) ) ) ) ) )
& ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9n: $i] : ( ap @ ( ap @ V2h @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V9n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V9n ) ) @ V0f ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V6l ) )
@ c_2Ereal__topology_2Esequentially ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V3k ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V10n: $i] : ( ap @ ( ap @ V2h @ ( ap @ V1g @ V10n ) ) @ ( ap @ V0f @ V10n ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__DIRICHLET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2N: tp__ty_2Enum_2Enum,V3k: tp__ty_2Enum_2Enum,V4m: tp__ty_2Enum_2Enum] :
( ( ( p
@ ( ap @ c_2Ereal__topology_2Ebounded__def
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ V5n ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V5n ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) )
& ! [V6n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2N ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum ) @ V1g ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ c_2Ereal__topology_2Esequentially ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( ap @ c_2Ereal__topology_2Efrom @ ( inj__ty_2Enum_2Enum @ V3k ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1g @ V7n ) ) @ ( ap @ V0f @ V7n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__INJECTIVE__IMAGE__STRONG,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V2f ) @ V1s ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0x @ V3n ) ) ) ) )
& ! [V4m: tp__ty_2Enum_2Enum,V5n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V1s ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V4m ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) )
=> ( V4m = V5n ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V2f ) @ V1s ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V6n ) ) ) @ V0x ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V1s ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V6n ) ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ V0x ) @ V2f ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__INJECTIVE__IMAGE,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V2f ) @ V1s ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0x @ V4n ) ) ) ) )
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ V1s ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V5m ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
=> ( V5m = V6n ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ V0x ) @ V2f ) ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V1s ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V2f ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__REARRANGE__EQ,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0x @ V4n ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epermutes @ ty_2Enum_2Enum ) @ V2p ) @ V1s ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ V0x ) @ V2p ) ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V1s ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESERIES__REARRANGE,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0x @ V4n ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epermutes @ ty_2Enum_2Enum ) @ V2p ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Ereal__topology_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ V0x ) @ V2p ) ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__REARRANGE,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0x @ V3n ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ereal__topology_2Epermutes @ ty_2Enum_2Enum ) @ V2p ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Esummable @ V1s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ V0x ) @ V2p ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBANACH__FIX,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ecomplete @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V1s ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) ) ) ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ V5x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDINI,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V2s ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V1g ) @ V2s ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V2s ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2E_2D_2D_3E @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ V0f @ V5n ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
@ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
@ c_2Ereal__topology_2Esequentially ) ) )
& ! [V6n: tp__ty_2Enum_2Enum,V7x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) ) @ ( ap @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) ) ) ) )
=> ! [V8e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V8e ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Eeventually @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9n: $i] :
( ap @ ( c_2Ebool_2E_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V10x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V10x ) @ V2s ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ V0f @ V9n ) @ V10x ) ) @ ( ap @ V1g @ V10x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V8e ) ) ) ) ) ) )
@ c_2Ereal__topology_2Esequentially ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Eclosest__point,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V0s ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V2x ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V3y ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__EXISTS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ V0s ) )
& ! [V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__IN__SET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__LE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__SELF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= V1x ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__REFL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= V1x )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EDIST__CLOSEST__POINT__LIPSCHITZ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__DIST__CLOSEST__POINT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ V2x ) ) ) ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__DIST__CLOSEST__POINT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ V2x ) ) ) ) )
@ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__DIST__CLOSEST__POINT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2x ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ V2x ) ) ) ) )
@ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEGMENT__TO__CLOSEST__POINT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEGMENT__TO__POINT__EXISTS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ V0s ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eopen__segment @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__IN__INTERIOR,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__IN__FRONTIER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2Einterior @ V0s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Esetdist,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ ( ap @ c_2Eiterate_2Einf
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2x ) @ V3y ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__EMPTY,axiom,
( ! [V0t: $i] :
( ( mem @ V0t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) @ V0t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__POS__LE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__SUBSETS__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2s_27: $i] :
( ( mem @ V2s_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2s_27 ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3t_27 ) @ V1t ) )
& ! [V4x: tp__ty_2Erealax_2Ereal,V5y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V1t ) ) )
=> ? [V6x_27: tp__ty_2Erealax_2Ereal,V7y_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x_27 ) ) @ V2s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7y_27 ) ) @ V3t_27 ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V7y_27 ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s_27 ) @ V3t_27 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LE__SETDIST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2d: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__LE__DIST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LE__SETDIST__EQ,axiom,
! [V0d: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0d ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V2t ) ) ) )
<=> ( ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0d ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) ) )
& ( ( ( V1s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( V2t
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0d ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__SETDIST__LT__EXISTS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ? [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1t ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__REFL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V0s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__SYM,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__TRIANGLE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V2t ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__SINGS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__LIPSCHITZ,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ A_27a )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__SETDIST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ ( c_2Ereal__topology_2Econtinuous @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V2y ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) )
@ ( ap @ c_2Ereal__topology_2Eat @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SETDIST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Econtinuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V2y ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) )
@ V1t ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SETDIST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p
@ ( ap
@ ( ap @ c_2Ereal__topology_2Euniformly__continuous__on
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] : ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V2y ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) )
@ V1t ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__DIFFERENCES,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ c_2Ereal__topology_2Esetdist
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__SUBSET__RIGHT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V2u ) ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__SUBSET__LEFT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V2u ) ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V2u ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__CLOSURE,axiom,
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V3t ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__FRONTIER,axiom,
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ ty_2Erealax_2Ereal ) @ V2s ) @ V3t ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V3t ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V3t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__COMPACT__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V1t ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__CLOSED__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V1t ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__COMPACT__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( V1t
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( V1t
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__BOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
| ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( V1t
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) )
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__TRANSLATION,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ c_2Ereal__topology_2Esetdist
@ ( ap
@ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) )
@ V1s ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V4x ) ) )
@ V2t ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2t ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V2t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__UNIQUE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) @ V1t ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
= V4d )
& ! [V5x: tp__ty_2Erealax_2Ereal,V6y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= V4d ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__UNIV,axiom,
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__ZERO,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__ZERO__STRONG,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__FRONTIERS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V1t ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__SING__FRONTIER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__CLOSEST__POINT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ c_2Ereal__topology_2Eclosest__point @ V1s ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__SING,axiom,
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V2s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED__IN,axiom,
! [V0u: $i] :
( ( mem @ V0u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0u ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0u ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V1s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__SING__IN__SET,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__SING__FRONTIER__CASES,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ c_2Ereal__topology_2Efrontier @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__SING__TRIANGLE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) ) ) @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__LE__SING,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__BALLS,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2r: tp__ty_2Erealax_2Ereal,V3s: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3s ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3s ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Emax @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V2r ) ) @ ( inj__ty_2Erealax_2Ereal @ V3s ) ) ) ) ) ) )
& ! [V4a: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal,V6r: tp__ty_2Erealax_2Ereal,V7s: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) @ ( inj__ty_2Erealax_2Ereal @ V7s ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V7s ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Emax @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V6r ) ) @ ( inj__ty_2Erealax_2Ereal @ V7s ) ) ) ) ) ) )
& ! [V8a: tp__ty_2Erealax_2Ereal,V9b: tp__ty_2Erealax_2Ereal,V10r: tp__ty_2Erealax_2Ereal,V11s: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V10r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Eball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) @ ( inj__ty_2Erealax_2Ereal @ V11s ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V10r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V11s ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Emax @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V10r ) ) @ ( inj__ty_2Erealax_2Ereal @ V11s ) ) ) ) ) ) )
& ! [V12a: tp__ty_2Erealax_2Ereal,V13b: tp__ty_2Erealax_2Ereal,V14r: tp__ty_2Erealax_2Ereal,V15s: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12a ) ) @ ( inj__ty_2Erealax_2Ereal @ V14r ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V13b ) ) @ ( inj__ty_2Erealax_2Ereal @ V15s ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V14r ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V15s ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Emax @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V12a ) ) @ ( inj__ty_2Erealax_2Ereal @ V13b ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V14r ) ) @ ( inj__ty_2Erealax_2Ereal @ V15s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEPARATION__CLOSURES,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1t ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V3v: $i] :
( ( mem @ V3v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ ty_2Erealax_2Ereal ) @ V2u ) @ V3v ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2u ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V3v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V3v ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEPARATION__NORMAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V3v: $i] :
( ( mem @ V3v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2u ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V3v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V3v ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2u ) @ V3v )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEPARATION__NORMAL__LOCAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V3s_27: $i] :
( ( mem @ V3s_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V4t_27: $i] :
( ( mem @ V4t_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V3s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V4t_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V3s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V4t_27 ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V3s_27 ) @ V4t_27 )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEPARATION__NORMAL__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V3v: $i] :
( ( mem @ V3v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2u ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ c_2Ereal__topology_2Eclosure @ V2u ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V3v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V3v ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2u ) @ V3v )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEPARATION__HAUSDORFF,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( V0x != V1y )
=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V3v: $i] :
( ( mem @ V3v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2u ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V3v ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ V3v ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2u ) @ V3v )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEPARATION__T2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( V0x != V1y )
<=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V3v: $i] :
( ( mem @ V3v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2u ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V3v ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ V3v ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2u ) @ V3v )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEPARATION__T1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( V0x != V1y )
<=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V3v: $i] :
( ( mem @ V3v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2u ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V3v ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2u ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ V2u ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V3v ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ V3v ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESEPARATION__T0,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( V0x != V1y )
<=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2u ) )
& ~ ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2u ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ V2u ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CHAIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0f ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V0f ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3t ) @ V0f ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2s ) @ V3t ) )
| ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3t ) @ V2s ) ) ) ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CHAIN__GEN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0f ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ V1s ) ) ) ) )
& ? [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V0f ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V2s ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V3s ) @ V0f ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V4t ) @ V0f ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) )
| ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4t ) @ V3s ) ) ) ) ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__NEST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ( ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( ap @ V0s @ V4n ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V4n ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONNECTED__NEST__GEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ( ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Econnected @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
& ? [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Econnected
@ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ ( ap @ V0s @ V5n ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ V5n ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Ehausdist,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal )
@ ( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ c_2Ebool_2E_7E
@ ( ap
@ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V3y ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) )
@ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4b: $i] :
( ap @ ( c_2Ebool_2E_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5d: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5d )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V6x ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6x ) @ V0s ) ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V7y ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7y ) @ V1t ) ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V5d ) @ V4b ) ) ) ) ) ) ) )
@ ( ap @ c_2Ereal_2Esup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V8x ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V8x ) @ V0s ) ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V9y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V9y ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V9y ) @ V1t ) ) ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__POS__LE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__REFL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V0s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__SYM,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__EMPTY,axiom,
( ! [V0t: $i] :
( ( mem @ V0t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) @ V0t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__SINGS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2s_27: $i] :
( ( mem @ V2s_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V4b: tp__ty_2Erealax_2Ereal] :
( ( ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) )
& ! [V6y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) ) )
<=> ( ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ V2s_27 ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V3t_27 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) )
& ! [V8y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8y ) ) @ V3t_27 ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V2s_27 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s_27 ) @ V3t_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__TRANSLATION,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ c_2Ereal__topology_2Ehausdist
@ ( ap
@ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) )
@ V1s ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V4x ) ) )
@ V2t ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__LINEAR__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V0f ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V2t ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V2t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__CLOSURE,axiom,
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V3t ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2s ) @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__HAUSDIST__LE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__HAUSDIST__LE__SUMS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p
@ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V4z: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V3y ) @ V4z ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4z ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ) ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5y: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V6z: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V5y ) @ V6z ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5y ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V6z ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LE__HAUSDIST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4c: tp__ty_2Erealax_2Ereal,V5z: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
& ! [V7y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7y ) ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4c ) ) ) )
& ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) ) )
| ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ V1t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__LE__HAUSDIST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__SING__LE__HAUSDIST,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESETDIST__HAUSDIST__TRIANGLE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V2u ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V2u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__SETDIST__TRIANGLE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V2u ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V2u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__LT__HAUSDIST__POINT__EXISTS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) ) )
=> ? [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1t ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EUPPER__LOWER__HEMICONTINUOUS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ V1t ) ) )
& ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V4u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V5x ) ) @ V4u ) ) ) ) ) ) ) ) )
& ! [V6u: $i] :
( ( mem @ V6u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V6u ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ V6u ) ) ) ) ) ) ) ) ) )
=> ! [V8x: tp__ty_2Erealax_2Ereal,V9e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) ) )
=> ? [V10d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V10d ) ) )
& ! [V11x_27: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V11x_27 ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ ( inj__ty_2Erealax_2Ereal @ V11x_27 ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V10d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V11x_27 ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V9e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__NONTRIVIAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ c_2Ereal_2Esup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V3y ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__NONTRIVIAL__ALT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Ereal_2Emax
@ ( ap @ c_2Ereal_2Esup
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) ) ) ) )
@ ( ap @ c_2Ereal_2Esup
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V3y ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EREAL__HAUSDIST__LE__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
<=> ( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
& ! [V4y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__UNION__LE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V2u ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V2u
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V2u ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V2u ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__INSERT__LE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ V1t ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__COMPACT__EXISTS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ? [V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V1t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__TRIANGLE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V2u ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V2u ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V2u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__COMPACT__SUMS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2y: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V3z: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V2y ) @ V3z ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2y ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3z ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__TRANS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V2u ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V2u ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) ) @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1t ) @ V2u ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__EQ__0,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( V1t
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
| ( ( ap @ c_2Ereal__topology_2Eclosure @ V0s )
= ( ap @ c_2Ereal__topology_2Eclosure @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__COMPACT__NONTRIVIAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1t ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ c_2Eiterate_2Einf
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2e: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2e )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2e ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V4y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V3x ) @ V4y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1t ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ V4y ) ) @ V2e ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V6y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ V5x ) @ V6y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V5x ) @ V0s ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ V6y ) ) @ V2e ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__ALT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V1t ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ c_2Ereal_2Esup
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V0s ) ) ) @ ( ap @ c_2Ereal__topology_2Esetdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ V1t ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DIAMETER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) )
=> ? [V2d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) )
& ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ebounded__def @ V3t ) )
& ( V3t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal__topology_2Ehausdist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0s ) @ V3t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal__topology_2Ediameter @ V0s ) ) @ ( ap @ c_2Ereal__topology_2Ediameter @ V3t ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EISOMETRY__IMP__OPEN__MAP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V4x: tp__ty_2Erealax_2Ereal,V5y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ V1s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V3u ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EISOMETRY__IMP__EMBEDDING,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s )
= V2t )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) @ V1s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) ) ) )
=> ? [V5g: $i] :
( ( mem @ V5g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V5g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EISOMETRY__IMP__HOMEOMORPHISM__COMPACT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V1s ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V1s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal__topology_2EDist @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) )
=> ? [V4g: $i] :
( ( mem @ V4g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V0f ) @ V4g ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EURYSOHN__LOCAL__STRONG,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal,V4b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V3a != V4b ) )
=> ? [V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V5f ) @ V2u ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ V2u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V5f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) )
& ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ V2u ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V5f @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
= V3a )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ V0s ) ) ) )
& ! [V8x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ V2u ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V5f @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) )
= V4b )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ V1t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EURYSOHN__LOCAL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3a: tp__ty_2Erealax_2Ereal,V4b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2u ) ) @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V5f ) @ V2u ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ V2u ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V5f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) ) )
& ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V5f @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
= V3a ) )
& ! [V8x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ V1t ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V5f @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) )
= V4b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EURYSOHN__STRONG,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V2a != V3b ) )
=> ? [V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V4f ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V4f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V4f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
= V2a )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ V0s ) ) )
& ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V4f @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
= V3b )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EURYSOHN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V4f ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V4f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) @ ( ap @ c_2Ereal__topology_2Eclosed__segment @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ) ) ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V4f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
= V2a ) )
& ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ V1t ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V4f @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
= V3b ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Elocally,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ V1s ) )
<=> ! [V2w: $i] :
( ( mem @ V2w @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V2w ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V2w ) ) )
=> ? [V4u: $i] :
( ( mem @ V4u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V5v: $i] :
( ( mem @ V5v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V4u ) )
& ( p @ ( ap @ V0P @ V5v ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V4u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4u ) @ V5v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5v ) @ V2w ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__MONO,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ V0P @ V3t ) )
=> ( p @ ( ap @ V1Q @ V3t ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ V2s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V1Q ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__OPEN__SUBSET,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__DIFF__CLOSED,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__EMPTY,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__SING,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
<=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__INTER,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ V1s ) )
& ( p @ ( ap @ V0P @ V2t ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1s ) @ V2t ) ) ) ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ V3s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ V4t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V3s ) @ V4t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__LOCALLY,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V5t: $i] :
( ( mem @ V5t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V4s ) @ V5t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V2f ) @ V3g ) ) )
=> ( ( p @ ( ap @ V0P @ V4s ) )
<=> ( p @ ( ap @ V1Q @ V5t ) ) ) ) ) )
=> ! [V6s: $i] :
( ( mem @ V6s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V7t: $i] :
( ( mem @ V7t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphism @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V6s ) @ V7t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) @ V2f ) @ V3g ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ V6s ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V1Q ) @ V7t ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__LOCALLY,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V2s ) @ V3t ) )
=> ( ( p @ ( ap @ V0P @ V2s ) )
<=> ( p @ ( ap @ V1Q @ V3t ) ) ) ) ) )
=> ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V5t: $i] :
( ( mem @ V5t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V4s ) @ V5t ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ V4s ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V1Q ) @ V5t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__TRANSLATION,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ! [V1a: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap @ V0P
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V3x ) ) )
@ V2s ) ) )
<=> ( p @ ( ap @ V0P @ V2s ) ) ) )
=> ! [V4a: tp__ty_2Erealax_2Ereal,V5s: $i] :
( ( mem @ V5s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ V6x ) ) )
@ V5s ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ V5s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V2f ) )
& ! [V4x: tp__ty_2Erealax_2Ereal,V5y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2f @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) )
=> ( V4x = V5y ) ) )
=> ( ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2f ) @ V3s ) ) )
<=> ( p @ ( ap @ V1Q @ V3s ) ) ) ) ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V7s: $i] :
( ( mem @ V7s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Elinear @ V6f ) )
& ! [V8x: tp__ty_2Erealax_2Ereal,V9y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V6f @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V6f @ ( inj__ty_2Erealax_2Ereal @ V9y ) ) ) )
=> ( V8x = V9y ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V6f ) @ V7s ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V1Q ) @ V7s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__OPEN__MAP__IMAGE,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V2f ) @ V3s ) )
& ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V3s ) ) @ V4t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2f ) @ V3s ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2f ) @ V4t ) ) ) ) )
& ! [V5t: $i] :
( ( mem @ V5t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V5t ) @ V3s ) )
& ( p @ ( ap @ V0P @ V5t ) ) )
=> ( p @ ( ap @ V1Q @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2f ) @ V5t ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V0P ) @ V3s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ V1Q ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V2f ) @ V3s ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Egdelta,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Egdelta @ V0s ) )
<=> ? [V1g: $i] :
( ( mem @ V1g @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1g ) )
& ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2u ) @ V1g ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2u ) ) ) )
& ( ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V1g )
= V0s ) ) ) ) ).
thf(ax_thm_2Ereal__topology_2Efsigma,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Efsigma @ V0s ) )
<=> ? [V1g: $i] :
( ( mem @ V1g @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1g ) )
& ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2c ) @ V1g ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V2c ) ) ) )
& ( ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V1g )
= V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EGDELTA__COMPLEMENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Egdelta @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Efsigma @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EFSIGMA__COMPLEMENT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Efsigma @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) @ V0s ) ) )
<=> ( p @ ( ap @ c_2Ereal__topology_2Egdelta @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__AS__GDELTA,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
=> ( p @ ( ap @ c_2Ereal__topology_2Egdelta @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V3v: $i] :
( ( mem @ V3v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2u ) @ V3v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3v ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V2u ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V3v ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__ALT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V2u ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ c_2Ereal__topology_2Eclosure @ V2u ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V2u ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__INTER__CBALL,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__INTER__CBALLS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
<=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ? [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ! [V3d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EClosed @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Ecball @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
<=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1k ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1k ) ) )
=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V3v: $i] :
( ( mem @ V3v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1k ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2u ) @ V3v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3v ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V2u ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V3v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__COMPACT__ALT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
<=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1k ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1k ) ) )
=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1k ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V2u ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ ( ap @ c_2Ereal__topology_2Eclosure @ V2u ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V2u ) ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__COMPACT__SUBOPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
<=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1k ) @ V0s ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V1k ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1k ) @ V2t ) ) )
=> ? [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V4v: $i] :
( ( mem @ V4v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1k ) @ V3u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3u ) @ V4v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3u ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V4v ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V3u ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V4v ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IMP__LOCALLY__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ECLOSED__IMP__LOCALLY__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__IMP__LOCALLY__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2Eis__interval @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__UNIV,axiom,
p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__INTER,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__OPEN__IN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__CLOSED__IN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__DELETE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Erealax_2Ereal ) @ V0s ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__LOCAL__COMPACTNESS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Ehomeomorphic @ V0s ) @ V1t ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__TRANSLATION__EQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V2x ) ) )
@ V1s ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V1s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__CLOSED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2EClosed ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__OPEN__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__CLOSED__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__LOCALLY__COMPACT,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V0s ) ) @ V1t ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) )
& ! [V2k: $i] :
( ( mem @ V2k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V2k ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2k ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2k ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V2k ) @ V1t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__PROPER__IMAGE__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ! [V2k: $i] :
( ( mem @ V2k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2k ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V2k ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V2k ) ) ) ) ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V1s ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__PROPER__IMAGE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ! [V2k: $i] :
( ( mem @ V2k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V2k ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V2k ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ V2k ) ) ) ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EMUMFORD__LEMMA,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ V0f ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V1s ) ) @ V2t ) )
& ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V2t ) )
& ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) ) ) )
=> ? [V5u: $i] :
( ( mem @ V5u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V6v: $i] :
( ( mem @ V6v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V5u ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V2t ) ) @ V6v ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V7x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V7x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) ) ) ) ) )
@ V5u ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V6v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0f ) @ V5u ) ) @ V6v ) )
& ! [V8k: $i] :
( ( mem @ V8k @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V8k ) @ V6v ) )
& ( p @ ( ap @ c_2Ereal__topology_2Ecompact @ V8k ) ) )
=> ( p
@ ( ap @ c_2Ereal__topology_2Ecompact
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V9x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V9x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V9x ) @ V5u ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V9x ) ) @ V8k ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__OPEN__INTER__CLOSURE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
=> ? [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) )
& ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1t ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__CLOSED__IN__OPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
=> ? [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1t ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ELOCALLY__COMPACT__CLOSED__INTER__OPEN,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V0s ) )
<=> ? [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1t ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2u ) )
& ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V1t ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBAIRE,axiom,
! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0g ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2t ) @ V0g ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1s ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V2t ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1s ) @ ( ap @ c_2Ereal__topology_2Eclosure @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ty_2Erealax_2Ereal ) @ V0g ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2EBAIRE__ALT,axiom,
! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Elocally @ c_2Ereal__topology_2Ecompact ) @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0g ) )
& ( ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0g )
= V1s ) )
=> ? [V2t: $i] :
( ( mem @ V2t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ? [V3u: $i] :
( ( mem @ V3u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2t ) @ V0g ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ereal__topology_2Esubtopology @ ty_2Erealax_2Ereal ) @ c_2Ereal__topology_2Eeuclidean ) @ V1s ) ) @ V3u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V3u ) @ ( ap @ c_2Ereal__topology_2Eclosure @ V2t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENOWHERE__DENSE__COUNTABLE__BIGUNION__CLOSED,axiom,
! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0g ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0g ) )
=> ( ( p @ ( ap @ c_2Ereal__topology_2EClosed @ V1s ) )
& ( ( ap @ c_2Ereal__topology_2Einterior @ V1s )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0g ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ENOWHERE__DENSE__COUNTABLE__BIGUNION,axiom,
! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0g ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ V0g ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ c_2Ereal__topology_2Eclosure @ V1s ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) )
=> ( ( ap @ c_2Ereal__topology_2Einterior @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0g ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(conj_thm_2Ereal__topology_2ESUBORDINATE__PARTITION__OF__UNITY,axiom,
! [V0c: $i] :
( ( mem @ V0c @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ty_2Erealax_2Ereal ) @ V0c ) ) )
& ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V2u ) @ V0c ) )
=> ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V2u ) ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1s ) )
=> ? [V4v: $i] :
( ( mem @ V4v @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V4v ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V4v ) )
& ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ( arr @ ty_2Erealax_2Ereal @ bool )
@ ^ [V5u: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ V5u ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V5u ) @ V0c ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Erealax_2Ereal ) @ V5u ) @ V4v ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ) ) ) ) )
=> ? [V6f: $i] :
( ( mem @ V6f @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
& ! [V7u: $i] :
( ( mem @ V7u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V7u ) @ V0c ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal__topology_2Econtinuous__on @ ( ap @ V6f @ V7u ) ) @ V1s ) )
& ! [V8x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ V6f @ V7u ) @ ( inj__ty_2Erealax_2Ereal @ V8x ) ) ) ) ) ) ) )
& ! [V9x: tp__ty_2Erealax_2Ereal,V10u: $i] :
( ( mem @ V10u @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V10u ) @ V0c ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) @ V1s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) @ V10u ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V6f @ V10u ) @ ( inj__ty_2Erealax_2Ereal @ V9x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
& ! [V11x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V11x ) ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V0c )
@ ( lam @ ( arr @ ty_2Erealax_2Ereal @ bool )
@ ^ [V12u: $i] : ( ap @ ( ap @ V6f @ V12u ) @ ( inj__ty_2Erealax_2Ereal @ V11x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ! [V13x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V13x ) ) @ V1s ) )
=> ? [V14n: $i] :
( ( mem @ V14n @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
& ( p @ ( ap @ c_2Ereal__topology_2EOpen @ V14n ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V13x ) ) @ V14n ) )
& ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
@ ( lam @ ( arr @ ty_2Erealax_2Ereal @ bool )
@ ^ [V15u: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ bool ) @ V15u )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V15u ) @ V0c ) )
@ ( ap @ c_2Ebool_2E_7E
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V16x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V16x ) @ V14n ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ V6f @ V15u ) @ V16x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------