ITP001 Axioms: ITP121^5.ax


%------------------------------------------------------------------------------
% File     : ITP121^5 : 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    : metric^2.ax [Gau20]
%          : HL4121^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   47 (   5 unt;  10 typ;   0 def)
%            Number of atoms       :  674 (  22 equ;   0 cnn)
%            Maximal formula atoms :   44 (  14 avg)
%            Number of connectives : 1050 (   2   ~;   0   |;   8   &; 980   @)
%                                         (   6 <=>;  54  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   32 (  11 avg; 980 nst)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :    8 (   8   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   44 (  43 usr;  35 con; 0-2 aty)
%            Number of variables   :  105 (  13   ^  89   !;   3   ?; 105   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Emetric_2Emetric,type,
    ty_2Emetric_2Emetric: del > del ).

thf(tp_c_2Emetric_2EB,type,
    c_2Emetric_2EB: del > $i ).

thf(mem_c_2Emetric_2EB,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Emetric_2EB @ A_27a ) @ ( arr @ ( ty_2Emetric_2Emetric @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Erealax_2Ereal ) @ ( arr @ A_27a @ bool ) ) ) ) ).

thf(tp_c_2Emetric_2Edist,type,
    c_2Emetric_2Edist: del > $i ).

thf(mem_c_2Emetric_2Edist,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Emetric_2Edist @ A_27a ) @ ( arr @ ( ty_2Emetric_2Emetric @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ty_2Erealax_2Ereal ) ) ) ).

thf(tp_c_2Emetric_2Eismet,type,
    c_2Emetric_2Eismet: del > $i ).

thf(mem_c_2Emetric_2Eismet,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Emetric_2Eismet @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ty_2Erealax_2Ereal ) @ bool ) ) ).

thf(tp_c_2Emetric_2Emetric,type,
    c_2Emetric_2Emetric: del > $i ).

thf(mem_c_2Emetric_2Emetric,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Emetric_2Emetric @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ty_2Erealax_2Ereal ) @ ( ty_2Emetric_2Emetric @ A_27a ) ) ) ).

thf(stp_c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal,type,
    tp__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal: $tType ).

thf(stp_inj_c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal,type,
    inj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal: tp__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal > $i ).

thf(stp_surj_c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal,type,
    surj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal: $i > tp__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal ).

thf(stp_inj_surj_c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal,axiom,
    ! [X: tp__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal] :
      ( ( surj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal @ ( inj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal @ X ) )
      = X ) ).

thf(stp_inj_mem_c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal,axiom,
    ! [X: tp__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal] : ( mem @ ( inj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal @ X ) @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) ) ).

thf(stp_iso_mem_c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal,axiom,
    ! [X: $i] :
      ( ( mem @ X @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) )
     => ( X
        = ( inj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal @ ( surj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal @ X ) ) ) ) ).

thf(tp_c_2Emetric_2Emr1,type,
    c_2Emetric_2Emr1: $i ).

thf(mem_c_2Emetric_2Emr1,axiom,
    mem @ c_2Emetric_2Emr1 @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) ).

thf(tp_c_2Emetric_2Emtop,type,
    c_2Emetric_2Emtop: del > $i ).

thf(mem_c_2Emetric_2Emtop,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Emetric_2Emtop @ A_27a ) @ ( arr @ ( ty_2Emetric_2Emetric @ A_27a ) @ ( ty_2Etopology_2Etopology @ A_27a ) ) ) ).

thf(ax_thm_2Emetric_2Eismet,axiom,
    ! [A_27a: del,V0m: $i] :
      ( ( mem @ V0m @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ty_2Erealax_2Ereal ) )
     => ( ( p @ ( ap @ ( c_2Emetric_2Eismet @ A_27a ) @ V0m ) )
      <=> ( ! [V1x: $i] :
              ( ( mem @ V1x @ A_27a )
             => ! [V2y: $i] :
                  ( ( mem @ V2y @ A_27a )
                 => ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0m @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) )
                      = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
                  <=> ( V1x = V2y ) ) ) )
          & ! [V3x: $i] :
              ( ( mem @ V3x @ A_27a )
             => ! [V4y: $i] :
                  ( ( mem @ V4y @ A_27a )
                 => ! [V5z: $i] :
                      ( ( mem @ V5z @ A_27a )
                     => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0m @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4y ) @ V5z ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0m @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3x ) @ V4y ) ) ) @ ( ap @ V0m @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3x ) @ V5z ) ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Emetric_2Emetric__TY__DEF,axiom,
    ! [A_27a: del] :
    ? [V0rep: $i] :
      ( ( mem @ V0rep @ ( arr @ ( ty_2Emetric_2Emetric @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ty_2Erealax_2Ereal ) ) )
      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ty_2Erealax_2Ereal ) @ ( ty_2Emetric_2Emetric @ A_27a ) ) @ ( c_2Emetric_2Eismet @ A_27a ) ) @ V0rep ) ) ) ).

thf(ax_thm_2Emetric_2Emetric__tybij,axiom,
    ! [A_27a: del] :
      ( ! [V0a: $i] :
          ( ( mem @ V0a @ ( ty_2Emetric_2Emetric @ A_27a ) )
         => ( ( ap @ ( c_2Emetric_2Emetric @ A_27a ) @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0a ) )
            = V0a ) )
      & ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ty_2Erealax_2Ereal ) )
         => ( ( p @ ( ap @ ( c_2Emetric_2Eismet @ A_27a ) @ V1r ) )
          <=> ( ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ ( ap @ ( c_2Emetric_2Emetric @ A_27a ) @ V1r ) )
              = V1r ) ) ) ) ).

thf(conj_thm_2Emetric_2EMETRIC__ISMET,axiom,
    ! [A_27a: del,V0m: $i] :
      ( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
     => ( p @ ( ap @ ( c_2Emetric_2Eismet @ A_27a ) @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) ) ) ) ).

thf(conj_thm_2Emetric_2EMETRIC__ZERO,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 )
             => ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) )
                  = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
              <=> ( V1x = V2y ) ) ) ) ) ).

thf(conj_thm_2Emetric_2EMETRIC__SAME,axiom,
    ! [A_27a: del,V0m: $i] :
      ( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V1x ) ) )
            = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).

thf(conj_thm_2Emetric_2EMETRIC__POS,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 )
             => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( 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 ) ) ) ) ) ) ) ).

thf(conj_thm_2Emetric_2EMETRIC__SYM,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 )
             => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) )
                = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2y ) @ V1x ) ) ) ) ) ) ) ).

thf(conj_thm_2Emetric_2EMETRIC__TRIANGLE,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 @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V3z ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( 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 ) @ V2y ) @ V3z ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Emetric_2EMETRIC__NZ,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 )
             => ( ( V1x != V2y )
               => ( 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 ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Emetric_2Emtop,axiom,
    ! [A_27a: del,V0m: $i] :
      ( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
     => ( ( ap @ ( c_2Emetric_2Emtop @ A_27a ) @ V0m )
        = ( ap @ ( c_2Etopology_2Etopology @ A_27a )
          @ ( lam @ ( arr @ A_27a @ bool )
            @ ^ [V1S_27: $i] :
                ( ap @ ( c_2Ebool_2E_21 @ A_27a )
                @ ( lam @ A_27a
                  @ ^ [V2x: $i] :
                      ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1S_27 @ V2x ) )
                      @ ( ap @ ( c_2Ebool_2E_3F @ ty_2Erealax_2Ereal )
                        @ ( lam @ ty_2Erealax_2Ereal
                          @ ^ [V3e: $i] :
                              ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V3e ) )
                              @ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
                                @ ( lam @ A_27a
                                  @ ^ [V4y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V4y ) ) ) @ V3e ) ) @ ( ap @ V1S_27 @ V4y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Emetric_2Emtop__istopology,axiom,
    ! [A_27a: del,V0m: $i] :
      ( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
     => ( p
        @ ( ap @ ( c_2Etopology_2Eistopology @ A_27a )
          @ ( lam @ ( arr @ A_27a @ bool )
            @ ^ [V1S_27: $i] :
                ( ap @ ( c_2Ebool_2E_21 @ A_27a )
                @ ( lam @ A_27a
                  @ ^ [V2x: $i] :
                      ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1S_27 @ V2x ) )
                      @ ( ap @ ( c_2Ebool_2E_3F @ ty_2Erealax_2Ereal )
                        @ ( lam @ ty_2Erealax_2Ereal
                          @ ^ [V3e: $i] :
                              ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V3e ) )
                              @ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
                                @ ( lam @ A_27a
                                  @ ^ [V4y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V4y ) ) ) @ V3e ) ) @ ( ap @ V1S_27 @ V4y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Emetric_2EMTOP__OPEN,axiom,
    ! [A_27a: del,V0S_27: $i] :
      ( ( mem @ V0S_27 @ ( arr @ A_27a @ bool ) )
     => ! [V1m: $i] :
          ( ( mem @ V1m @ ( ty_2Emetric_2Emetric @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ ( ap @ ( c_2Emetric_2Emtop @ A_27a ) @ V1m ) ) @ V0S_27 ) )
          <=> ! [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
               => ( ( p @ ( ap @ V0S_27 @ V2x ) )
                 => ? [V3e: tp__ty_2Erealax_2Ereal] :
                      ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
                      & ! [V4y: $i] :
                          ( ( mem @ V4y @ A_27a )
                         => ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V1m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V4y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
                           => ( p @ ( ap @ V0S_27 @ V4y ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Emetric_2Eball,axiom,
    ! [A_27a: del,V0m: $i] :
      ( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2e: tp__ty_2Erealax_2Ereal] :
              ( ( ap @ ( ap @ ( c_2Emetric_2EB @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Erealax_2Ereal ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
              = ( lam @ A_27a
                @ ^ [V3y: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V3y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ).

thf(conj_thm_2Emetric_2EBALL__OPEN,axiom,
    ! [A_27a: del,V0m: $i] :
      ( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2e: tp__ty_2Erealax_2Ereal] :
              ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
             => ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ ( ap @ ( c_2Emetric_2Emtop @ A_27a ) @ V0m ) ) @ ( ap @ ( ap @ ( c_2Emetric_2EB @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Erealax_2Ereal ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Emetric_2EBALL__NEIGH,axiom,
    ! [A_27a: del,V0m: $i] :
      ( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2e: tp__ty_2Erealax_2Ereal] :
              ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
             => ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eneigh @ A_27a ) @ ( ap @ ( c_2Emetric_2Emtop @ A_27a ) @ V0m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ A_27a ) @ ( ap @ ( ap @ ( c_2Emetric_2EB @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Erealax_2Ereal ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) @ V1x ) ) ) ) ) ) ).

thf(conj_thm_2Emetric_2EMTOP__LIMPT,axiom,
    ! [A_27a: del,V0m: $i] :
      ( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2S_27: $i] :
              ( ( mem @ V2S_27 @ ( arr @ A_27a @ bool ) )
             => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etopology_2Elimpt @ A_27a ) @ ( ap @ ( c_2Emetric_2Emtop @ A_27a ) @ V0m ) ) @ V1x ) @ V2S_27 ) )
              <=> ! [V3e: tp__ty_2Erealax_2Ereal] :
                    ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
                   => ? [V4y: $i] :
                        ( ( mem @ V4y @ A_27a )
                        & ( V1x != V4y )
                        & ( p @ ( ap @ V2S_27 @ V4y ) )
                        & ( 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 ) @ V1x ) @ V4y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Emetric_2EISMET__R1,axiom,
    ( p
    @ ( ap @ ( c_2Emetric_2Eismet @ ty_2Erealax_2Ereal )
      @ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
        @ ( lam @ ty_2Erealax_2Ereal
          @ ^ [V0x: $i] :
              ( lam @ ty_2Erealax_2Ereal
              @ ^ [V1y: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ V1y ) @ V0x ) ) ) ) ) ) ) ).

thf(ax_thm_2Emetric_2Emr1,axiom,
    ( ( surj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal @ c_2Emetric_2Emr1 )
    = ( surj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal
      @ ( ap @ ( c_2Emetric_2Emetric @ ty_2Erealax_2Ereal )
        @ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
          @ ( lam @ ty_2Erealax_2Ereal
            @ ^ [V0x: $i] :
                ( lam @ ty_2Erealax_2Ereal
                @ ^ [V1y: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ V1y ) @ V0x ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Emetric_2EMR1__DEF,axiom,
    ! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
      ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) @ ( 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 @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).

thf(conj_thm_2Emetric_2EMR1__ADD,axiom,
    ! [V0x: tp__ty_2Erealax_2Ereal,V1d: tp__ty_2Erealax_2Ereal] :
      ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
      = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) ).

thf(conj_thm_2Emetric_2EMR1__SUB,axiom,
    ! [V0x: tp__ty_2Erealax_2Ereal,V1d: tp__ty_2Erealax_2Ereal] :
      ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
      = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) ).

thf(conj_thm_2Emetric_2EMR1__ADD__POS,axiom,
    ! [V0x: tp__ty_2Erealax_2Ereal,V1d: 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 @ V1d ) ) )
     => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
        = V1d ) ) ).

thf(conj_thm_2Emetric_2EMR1__SUB__LE,axiom,
    ! [V0x: tp__ty_2Erealax_2Ereal,V1d: 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 @ V1d ) ) )
     => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
        = V1d ) ) ).

thf(conj_thm_2Emetric_2EMR1__ADD__LT,axiom,
    ! [V0x: tp__ty_2Erealax_2Ereal,V1d: 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 @ V1d ) ) )
     => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
        = V1d ) ) ).

thf(conj_thm_2Emetric_2EMR1__SUB__LT,axiom,
    ! [V0x: tp__ty_2Erealax_2Ereal,V1d: 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 @ V1d ) ) )
     => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
        = V1d ) ) ).

thf(conj_thm_2Emetric_2EMR1__BETWEEN1,axiom,
    ! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
      ( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) )
        & ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) )
     => ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ).

thf(conj_thm_2Emetric_2EMR1__LIMPT,axiom,
    ! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etopology_2Elimpt @ ty_2Erealax_2Ereal ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ).

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