ITP001 Axioms: ITP124^5.ax
%------------------------------------------------------------------------------
% File : ITP124^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : nets^2.ax [Gau20]
% : HL4124^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 42 ( 1 unt; 7 typ; 0 def)
% Number of atoms : 1539 ( 8 equ; 0 cnn)
% Maximal formula atoms : 90 ( 36 avg)
% Number of connectives : 2401 ( 5 ~; 0 |; 43 &;2201 @)
% ( 11 <=>; 141 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 20 avg;2201 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 8 ( 8 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 42 usr; 36 con; 0-2 aty)
% Number of variables : 186 ( 12 ^ 158 !; 16 ?; 186 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Enets_2Ebounded,type,
c_2Enets_2Ebounded: del > del > $i ).
thf(mem_c_2Enets_2Ebounded,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Enets_2Ebounded @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ ( ty_2Emetric_2Emetric @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) ) @ ( arr @ ( arr @ A_27b @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Enets_2Edorder,type,
c_2Enets_2Edorder: del > $i ).
thf(mem_c_2Enets_2Edorder,axiom,
! [A_27a: del] : ( mem @ ( c_2Enets_2Edorder @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Enets_2Etends,type,
c_2Enets_2Etends: del > del > $i ).
thf(mem_c_2Enets_2Etends,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Enets_2Etends @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Epair_2Eprod @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) ) @ bool ) ) ) ) ).
thf(tp_c_2Enets_2Etendsto,type,
c_2Enets_2Etendsto: del > $i ).
thf(mem_c_2Enets_2Etendsto,axiom,
! [A_27a: del] : ( mem @ ( c_2Enets_2Etendsto @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( ty_2Emetric_2Emetric @ A_27a ) @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(ax_thm_2Enets_2Edorder,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27a ) @ V0g ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0g @ V1x ) @ V1x ) )
& ( p @ ( ap @ ( ap @ V0g @ V2y ) @ V2y ) ) )
=> ? [V3z: $i] :
( ( mem @ V3z @ A_27a )
& ( p @ ( ap @ ( ap @ V0g @ V3z ) @ V3z ) )
& ! [V4w: $i] :
( ( mem @ V4w @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0g @ V4w ) @ V3z ) )
=> ( ( p @ ( ap @ ( ap @ V0g @ V4w ) @ V1x ) )
& ( p @ ( ap @ ( ap @ V0g @ V4w ) @ V2y ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Enets_2Etends,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27b @ A_27a ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ A_27a )
=> ! [V2top: $i] :
( ( mem @ V2top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ A_27a @ A_27b ) @ V0s ) @ V1l ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) ) @ V2top ) @ V3g ) ) )
<=> ! [V4N: $i] :
( ( mem @ V4N @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eneigh @ A_27a ) @ V2top ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ A_27a ) @ V4N ) @ V1l ) ) )
=> ? [V5n: $i] :
( ( mem @ V5n @ A_27b )
& ( p @ ( ap @ ( ap @ V3g @ V5n ) @ V5n ) )
& ! [V6m: $i] :
( ( mem @ V6m @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V3g @ V6m ) @ V5n ) )
=> ( p @ ( ap @ V4N @ ( ap @ V0s @ V6m ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Enets_2Ebounded,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) ) @ V0m ) @ V1g ) ) @ V2f ) )
<=> ? [V3k: tp__ty_2Erealax_2Ereal,V4x: $i] :
( ( mem @ V4x @ A_27a )
& ? [V5N: $i] :
( ( mem @ V5N @ A_27b )
& ( p @ ( ap @ ( ap @ V1g @ V5N ) @ V5N ) )
& ! [V6n: $i] :
( ( mem @ V6n @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V1g @ V6n ) @ V5N ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ ( ap @ V2f @ V6n ) ) @ V4x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3k ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Enets_2Etendsto,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etendsto @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ A_27a ) @ A_27a ) @ V0m ) @ V1x ) ) @ V2y ) @ V3z ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) ) @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V3z ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2EDORDER__LEMMA,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27a ) @ V0g ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ( ? [V3n: $i] :
( ( mem @ V3n @ A_27a )
& ( p @ ( ap @ ( ap @ V0g @ V3n ) @ V3n ) )
& ! [V4m: $i] :
( ( mem @ V4m @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0g @ V4m ) @ V3n ) )
=> ( p @ ( ap @ V1P @ V4m ) ) ) ) )
& ? [V5n: $i] :
( ( mem @ V5n @ A_27a )
& ( p @ ( ap @ ( ap @ V0g @ V5n ) @ V5n ) )
& ! [V6m: $i] :
( ( mem @ V6m @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0g @ V6m ) @ V5n ) )
=> ( p @ ( ap @ V2Q @ V6m ) ) ) ) ) )
=> ? [V7n: $i] :
( ( mem @ V7n @ A_27a )
& ( p @ ( ap @ ( ap @ V0g @ V7n ) @ V7n ) )
& ! [V8m: $i] :
( ( mem @ V8m @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0g @ V8m ) @ V7n ) )
=> ( ( p @ ( ap @ V1P @ V8m ) )
& ( p @ ( ap @ V2Q @ V8m ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2EDORDER__NGE,axiom,
p @ ( ap @ ( c_2Enets_2Edorder @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_3E_3D ) ).
thf(conj_thm_2Enets_2EDORDER__TENDSTO,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27a ) @ ( ap @ ( c_2Enets_2Etendsto @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ A_27a ) @ A_27a ) @ V0m ) @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2EMTOP__TENDS,axiom,
! [A_27a: del,A_27b: del,V0d: $i] :
( ( mem @ V0d @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ A_27b @ A_27a ) )
=> ! [V3x0: $i] :
( ( mem @ V3x0 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ A_27a @ A_27b ) @ V2x ) @ V3x0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ A_27a ) @ V0d ) ) @ V1g ) ) )
<=> ! [V4e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
=> ? [V5n: $i] :
( ( mem @ V5n @ A_27b )
& ( p @ ( ap @ ( ap @ V1g @ V5n ) @ V5n ) )
& ! [V6m: $i] :
( ( mem @ V6m @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V1g @ V6m ) @ V5n ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0d ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ ( ap @ V2x @ V6m ) ) @ V3x0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2EMTOP__TENDS__UNIQ,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27b @ A_27a ) )
=> ! [V1x0: $i] :
( ( mem @ V1x0 @ A_27a )
=> ! [V2x1: $i] :
( ( mem @ V2x1 @ A_27a )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4d: $i] :
( ( mem @ V4d @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27b ) @ V3g ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ A_27a @ A_27b ) @ V0x ) @ V1x0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ A_27a ) @ V4d ) ) @ V3g ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ A_27a @ A_27b ) @ V0x ) @ V2x1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ A_27a ) @ V4d ) ) @ V3g ) ) ) )
=> ( V1x0 = V2x1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ESEQ__TENDS,axiom,
! [A_27a: del,V0d: $i] :
( ( mem @ V0d @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2x0: $i] :
( ( mem @ V2x0 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ A_27a @ ty_2Enum_2Enum ) @ V1x ) @ V2x0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ A_27a ) @ V0d ) ) @ c_2Earithmetic_2E_3E_3D ) ) )
<=> ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4N: tp__ty_2Enum_2Enum] :
! [V5n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ ( inj__ty_2Enum_2Enum @ V4N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0d ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ ( ap @ V1x @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) @ V2x0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ELIM__TENDS,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Emetric_2Emetric @ A_27b ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3x0: $i] :
( ( mem @ V3x0 @ A_27a )
=> ! [V4y0: $i] :
( ( mem @ V4y0 @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etopology_2Elimpt @ A_27a ) @ ( ap @ ( c_2Emetric_2Emtop @ A_27a ) @ V0m1 ) ) @ V3x0 ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ A_27b @ A_27a ) @ V2f ) @ V4y0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ A_27b ) @ V1m2 ) ) @ ( ap @ ( c_2Enets_2Etendsto @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ A_27a ) @ A_27a ) @ V0m1 ) @ V3x0 ) ) ) ) )
<=> ! [V5e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) )
=> ? [V6d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6d ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V3x0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V3x0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27b ) @ V1m2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27b ) @ ( ap @ V2f @ V7x ) ) @ V4y0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ELIM__TENDS2,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Emetric_2Emetric @ A_27b ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3x0: $i] :
( ( mem @ V3x0 @ A_27a )
=> ! [V4y0: $i] :
( ( mem @ V4y0 @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etopology_2Elimpt @ A_27a ) @ ( ap @ ( c_2Emetric_2Emtop @ A_27a ) @ V0m1 ) ) @ V3x0 ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ A_27b @ A_27a ) @ V2f ) @ V4y0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ A_27b ) @ V1m2 ) ) @ ( ap @ ( c_2Enets_2Etendsto @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ A_27a ) @ A_27a ) @ V0m1 ) @ V3x0 ) ) ) ) )
<=> ! [V5e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) )
=> ? [V6d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6d ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V3x0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V3x0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27b ) @ V1m2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27b ) @ ( ap @ V2f @ V7x ) ) @ V4y0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2EMR1__BOUNDED,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ V0g ) ) @ V1f ) )
<=> ? [V2k: tp__ty_2Erealax_2Ereal,V3N: $i] :
( ( mem @ V3N @ A_27a )
& ( p @ ( ap @ ( ap @ V0g @ V3N ) @ V3N ) )
& ! [V4n: $i] :
( ( mem @ V4n @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0g @ V4n ) @ V3N ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V1f @ V4n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2k ) ) ) ) ) ) ) ) ) ).
thf(stp_c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal,type,
tp__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal: $tType ).
thf(stp_inj_c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal,type,
inj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal: tp__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal > $i ).
thf(stp_surj_c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal,type,
surj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal: $i > tp__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal ).
thf(stp_inj_surj_c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal @ ( inj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal] : ( mem @ ( inj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal @ X ) @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) ) ).
thf(stp_iso_mem_c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) )
=> ( X
= ( inj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal @ ( surj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal @ X ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__NULL,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1x @ V3n ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__CONV__BOUNDED,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ V0g ) ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__CONV__NZ,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
& ( V2x0
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ? [V3N: $i] :
( ( mem @ V3N @ A_27a )
& ( p @ ( ap @ ( ap @ V0g @ V3N ) @ V3N ) )
& ! [V4n: $i] :
( ( mem @ V4n @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0g @ V4n ) @ V3N ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1x @ V4n ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__CONV__IBOUNDED,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
& ( V2x0
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ V0g ) )
@ ( lam @ A_27a
@ ^ [V3n: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ V1x @ V3n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__NULL__ADD,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27a ) @ V0g ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V2y ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V1x @ V3n ) ) @ ( ap @ V2y @ V3n ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__NULL__MUL,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27a ) @ V0g ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ V0g ) ) @ V1x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V2y ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1x @ V3n ) ) @ ( ap @ V2y @ V3n ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__NULL__CMUL,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1k: tp__ty_2Erealax_2Ereal,V2x: $i] :
( ( mem @ V2x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V2x ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1k ) ) @ ( ap @ V2x @ V3n ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__ADD,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27a ) @ V0g ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2x0: tp__ty_2Erealax_2Ereal,V3y: $i] :
( ( mem @ V3y @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V4y0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V3y ) @ ( inj__ty_2Erealax_2Ereal @ V4y0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V5n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V1x @ V5n ) ) @ ( ap @ V3y @ V5n ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y0 ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__NEG,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27a ) @ V0g ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
<=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V3n: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V1x @ V3n ) ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__SUB,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27a ) @ V0g ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2x0: tp__ty_2Erealax_2Ereal,V3y: $i] :
( ( mem @ V3y @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V4y0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V3y ) @ ( inj__ty_2Erealax_2Ereal @ V4y0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V5n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1x @ V5n ) ) @ ( ap @ V3y @ V5n ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y0 ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__MUL,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27a ) @ V0g ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3x0: tp__ty_2Erealax_2Ereal,V4y0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V3x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V2y ) @ ( inj__ty_2Erealax_2Ereal @ V4y0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V5n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1x @ V5n ) ) @ ( ap @ V2y @ V5n ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y0 ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__INV,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27a ) @ V0g ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
& ( V2x0
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V3n: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ V1x @ V3n ) ) ) )
@ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__DIV,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27a ) @ V0g ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2x0: tp__ty_2Erealax_2Ereal,V3y: $i] :
( ( mem @ V3y @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V4y0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V3y ) @ ( inj__ty_2Erealax_2Ereal @ V4y0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
& ( V4y0
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V5n: $i] : ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ V1x @ V5n ) ) @ ( ap @ V3y @ V5n ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y0 ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__ABS,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a )
@ ( lam @ A_27a
@ ^ [V3n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V1x @ V3n ) ) ) )
@ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) ) ) ) ) ).
thf(conj_thm_2Enets_2ENET__LE,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Enets_2Edorder @ A_27a ) @ V0g ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2x0: tp__ty_2Erealax_2Ereal,V3y: $i] :
( ( mem @ V3y @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V4y0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ A_27a ) @ V3y ) @ ( inj__ty_2Erealax_2Ereal @ V4y0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ V0g ) ) )
& ? [V5N: $i] :
( ( mem @ V5N @ A_27a )
& ( p @ ( ap @ ( ap @ V0g @ V5N ) @ V5N ) )
& ! [V6n: $i] :
( ( mem @ V6n @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0g @ V6n ) @ V5N ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1x @ V6n ) ) @ ( ap @ V3y @ V6n ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y0 ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------