TPTP Problem File: ITP022^5.p
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%------------------------------------------------------------------------------
% File : ITP022^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Ereal__topology_2EINDEPENDENT__STDBASIS.p, 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 : thm_2Ereal__topology_2EINDEPENDENT__STDBASIS.p [Gau20]
% : HL410501^5.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 20394 (2672 unt;3305 typ; 0 def)
% Number of atoms : 471287 (18289 equ; 0 cnn)
% Maximal formula atoms : 9798 ( 27 avg)
% Number of connectives : 856399 (2576 ~;1066 |;26549 &;786030 @)
% (6282 <=>;33896 =>; 0 <=; 0 <~>)
% Maximal formula depth : 368 ( 11 avg)
% Number of types : 153 ( 151 usr)
% Number of type conns : 2880 (2880 >; 0 *; 0 +; 0 <<)
% Number of symbols : 3229 (3226 usr;1177 con; 0-6 aty)
% Number of variables : 74564 (4749 ^;55866 !;13949 ?;74564 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
include('Axioms/ITP001/ITP002^5.ax').
include('Axioms/ITP001/ITP003^5.ax').
include('Axioms/ITP001/ITP004^5.ax').
include('Axioms/ITP001/ITP007^5.ax').
include('Axioms/ITP001/ITP006^5.ax').
include('Axioms/ITP001/ITP005^5.ax').
include('Axioms/ITP001/ITP008^5.ax').
include('Axioms/ITP001/ITP009^5.ax').
include('Axioms/ITP001/ITP010^5.ax').
include('Axioms/ITP001/ITP012^5.ax').
include('Axioms/ITP001/ITP011^5.ax').
include('Axioms/ITP001/ITP013^5.ax').
include('Axioms/ITP001/ITP014^5.ax').
include('Axioms/ITP001/ITP015^5.ax').
include('Axioms/ITP001/ITP017^5.ax').
include('Axioms/ITP001/ITP016^5.ax').
include('Axioms/ITP001/ITP019^5.ax').
include('Axioms/ITP001/ITP018^5.ax').
include('Axioms/ITP001/ITP021^5.ax').
include('Axioms/ITP001/ITP022^5.ax').
include('Axioms/ITP001/ITP020^5.ax').
include('Axioms/ITP001/ITP024^5.ax').
include('Axioms/ITP001/ITP023^5.ax').
include('Axioms/ITP001/ITP025^5.ax').
include('Axioms/ITP001/ITP026^5.ax').
include('Axioms/ITP001/ITP027^5.ax').
include('Axioms/ITP001/ITP028^5.ax').
include('Axioms/ITP001/ITP031^5.ax').
include('Axioms/ITP001/ITP029^5.ax').
include('Axioms/ITP001/ITP033^5.ax').
include('Axioms/ITP001/ITP030^5.ax').
include('Axioms/ITP001/ITP032^5.ax').
include('Axioms/ITP001/ITP038^5.ax').
include('Axioms/ITP001/ITP035^5.ax').
include('Axioms/ITP001/ITP034^5.ax').
include('Axioms/ITP001/ITP036^5.ax').
include('Axioms/ITP001/ITP037^5.ax').
include('Axioms/ITP001/ITP039^5.ax').
include('Axioms/ITP001/ITP041^5.ax').
include('Axioms/ITP001/ITP042^5.ax').
include('Axioms/ITP001/ITP040^5.ax').
include('Axioms/ITP001/ITP044^5.ax').
include('Axioms/ITP001/ITP051^5.ax').
include('Axioms/ITP001/ITP045^5.ax').
include('Axioms/ITP001/ITP056^5.ax').
include('Axioms/ITP001/ITP046^5.ax').
include('Axioms/ITP001/ITP043^5.ax').
include('Axioms/ITP001/ITP052^5.ax').
include('Axioms/ITP001/ITP057^5.ax').
include('Axioms/ITP001/ITP048^5.ax').
include('Axioms/ITP001/ITP047^5.ax').
include('Axioms/ITP001/ITP055^5.ax').
include('Axioms/ITP001/ITP053^5.ax').
include('Axioms/ITP001/ITP054^5.ax').
include('Axioms/ITP001/ITP058^5.ax').
include('Axioms/ITP001/ITP049^5.ax').
include('Axioms/ITP001/ITP050^5.ax').
include('Axioms/ITP001/ITP061^5.ax').
include('Axioms/ITP001/ITP069^5.ax').
include('Axioms/ITP001/ITP062^5.ax').
include('Axioms/ITP001/ITP068^5.ax').
include('Axioms/ITP001/ITP078^5.ax').
include('Axioms/ITP001/ITP064^5.ax').
include('Axioms/ITP001/ITP060^5.ax').
include('Axioms/ITP001/ITP067^5.ax').
include('Axioms/ITP001/ITP075^5.ax').
include('Axioms/ITP001/ITP074^5.ax').
include('Axioms/ITP001/ITP063^5.ax').
include('Axioms/ITP001/ITP059^5.ax').
include('Axioms/ITP001/ITP065^5.ax').
include('Axioms/ITP001/ITP076^5.ax').
include('Axioms/ITP001/ITP066^5.ax').
include('Axioms/ITP001/ITP077^5.ax').
include('Axioms/ITP001/ITP070^5.ax').
include('Axioms/ITP001/ITP071^5.ax').
include('Axioms/ITP001/ITP072^5.ax').
include('Axioms/ITP001/ITP073^5.ax').
include('Axioms/ITP001/ITP080^5.ax').
include('Axioms/ITP001/ITP083^5.ax').
include('Axioms/ITP001/ITP088^5.ax').
include('Axioms/ITP001/ITP082^5.ax').
include('Axioms/ITP001/ITP081^5.ax').
include('Axioms/ITP001/ITP079^5.ax').
include('Axioms/ITP001/ITP087^5.ax').
include('Axioms/ITP001/ITP085^5.ax').
include('Axioms/ITP001/ITP086^5.ax').
include('Axioms/ITP001/ITP090^5.ax').
include('Axioms/ITP001/ITP089^5.ax').
include('Axioms/ITP001/ITP084^5.ax').
include('Axioms/ITP001/ITP104^5.ax').
include('Axioms/ITP001/ITP097^5.ax').
include('Axioms/ITP001/ITP094^5.ax').
include('Axioms/ITP001/ITP095^5.ax').
include('Axioms/ITP001/ITP093^5.ax').
include('Axioms/ITP001/ITP102^5.ax').
include('Axioms/ITP001/ITP091^5.ax').
include('Axioms/ITP001/ITP098^5.ax').
include('Axioms/ITP001/ITP096^5.ax').
include('Axioms/ITP001/ITP100^5.ax').
include('Axioms/ITP001/ITP092^5.ax').
include('Axioms/ITP001/ITP105^5.ax').
include('Axioms/ITP001/ITP101^5.ax').
include('Axioms/ITP001/ITP103^5.ax').
include('Axioms/ITP001/ITP099^5.ax').
include('Axioms/ITP001/ITP111^5.ax').
include('Axioms/ITP001/ITP108^5.ax').
include('Axioms/ITP001/ITP106^5.ax').
include('Axioms/ITP001/ITP107^5.ax').
include('Axioms/ITP001/ITP114^5.ax').
include('Axioms/ITP001/ITP110^5.ax').
include('Axioms/ITP001/ITP109^5.ax').
include('Axioms/ITP001/ITP112^5.ax').
include('Axioms/ITP001/ITP113^5.ax').
include('Axioms/ITP001/ITP117^5.ax').
include('Axioms/ITP001/ITP116^5.ax').
include('Axioms/ITP001/ITP118^5.ax').
include('Axioms/ITP001/ITP120^5.ax').
include('Axioms/ITP001/ITP119^5.ax').
include('Axioms/ITP001/ITP121^5.ax').
include('Axioms/ITP001/ITP123^5.ax').
include('Axioms/ITP001/ITP124^5.ax').
include('Axioms/ITP001/ITP126^5.ax').
include('Axioms/ITP001/ITP127^5.ax').
include('Axioms/ITP001/ITP129^5.ax').
include('Axioms/ITP001/ITP128^5.ax').
include('Axioms/ITP001/ITP130^5.ax').
include('Axioms/ITP001/ITP131^5.ax').
include('Axioms/ITP001/ITP132^5.ax').
include('Axioms/ITP001/ITP134^5.ax').
include('Axioms/ITP001/ITP133^5.ax').
include('Axioms/ITP001/ITP135^5.ax').
include('Axioms/ITP001/ITP136^5.ax').
include('Axioms/ITP001/ITP138^5.ax').
include('Axioms/ITP001/ITP137^5.ax').
include('Axioms/ITP001/ITP140^5.ax').
include('Axioms/ITP001/ITP141^5.ax').
include('Axioms/ITP001/ITP139^5.ax').
include('Axioms/ITP001/ITP142^5.ax').
include('Axioms/ITP001/ITP145^5.ax').
include('Axioms/ITP001/ITP144^5.ax').
include('Axioms/ITP001/ITP143^5.ax').
include('Axioms/ITP001/ITP147^5.ax').
include('Axioms/ITP001/ITP146^5.ax').
%------------------------------------------------------------------------------
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,conjecture,
( 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 ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------