ITP001 Axioms: ITP124^5.ax


%------------------------------------------------------------------------------
% File     : ITP124^5 : TPTP v9.0.0. 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 ) ) ) ) ) ) ) ) ).

%------------------------------------------------------------------------------