%------------------------------------------------------------------------------
% File     : ITP003^3 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Problem  : HOL4 syntactic export of thm_2Earithmetic_2EMOD__2.p, bushy mode
% Version  : [BG+19] axioms.
% English  : 

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : thm_2Earithmetic_2EMOD__2.p [Gau19]
%          : HL401001^3.p [TPAP]

% Status   : Theorem
% Rating   : 0.67 v8.1.0, 0.50 v7.5.0
% Syntax   : Number of formulae    :   90 (  21 unt;  25 typ;   0 def)
%            Number of atoms       :  149 (  27 equ;  59 cnn)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives :  451 (  59   ~;  53   |;  46   &; 194   @)
%                                         (  62 <=>;  37  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   11 (   6 avg)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   50 (  50   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   25 (  23 usr;   5 con; 0-4 aty)
%            Number of variables   :  157 (   0   ^; 135   !;  18   ?; 157   :)
%                                         (   4  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

% Comments : 
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

thf(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: $tType > $tType > $tType ).

thf(tyop_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $tType ).

thf(c_2Ebool_2E_21,type,
    c_2Ebool_2E_21: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Earithmetic_2E_2A,type,
    c_2Earithmetic_2E_2A: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Earithmetic_2E_2B,type,
    c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Ebool_2E_2F_5C,type,
    c_2Ebool_2E_2F_5C: $o > $o > $o ).

thf(c_2Enum_2E0,type,
    c_2Enum_2E0: tyop_2Enum_2Enum ).

thf(c_2Eprim__rec_2E_3C,type,
    c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).

thf(c_2Emin_2E_3D,type,
    c_2Emin_2E_3D: 
      !>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).

thf(c_2Emin_2E_3D_3D_3E,type,
    c_2Emin_2E_3D_3D_3E: $o > $o > $o ).

thf(c_2Ebool_2E_3F,type,
    c_2Ebool_2E_3F: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Earithmetic_2EBIT1,type,
    c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Earithmetic_2EBIT2,type,
    c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Ebool_2ECOND,type,
    c_2Ebool_2ECOND: 
      !>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).

thf(c_2Earithmetic_2EEVEN,type,
    c_2Earithmetic_2EEVEN: tyop_2Enum_2Enum > $o ).

thf(c_2Ebool_2EF,type,
    c_2Ebool_2EF: $o ).

thf(c_2Earithmetic_2EMOD,type,
    c_2Earithmetic_2EMOD: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Earithmetic_2ENUMERAL,type,
    c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Earithmetic_2EODD,type,
    c_2Earithmetic_2EODD: tyop_2Enum_2Enum > $o ).

thf(c_2Enum_2ESUC,type,
    c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Ebool_2ET,type,
    c_2Ebool_2ET: $o ).

thf(c_2Earithmetic_2EZERO,type,
    c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).

thf(c_2Ebool_2E_5C_2F,type,
    c_2Ebool_2E_5C_2F: $o > $o > $o ).

thf(c_2Ebool_2E_7E,type,
    c_2Ebool_2E_7E: $o > $o ).

thf(logicdef_2E_2F_5C,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
    <=> ( V0
        & V1 ) ) ).

thf(logicdef_2E_5C_2F,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
    <=> ( V0
        | V1 ) ) ).

thf(logicdef_2E_7E,axiom,
    ! [V0: $o] :
      ( ( c_2Ebool_2E_7E @ V0 )
    <=> ( (~) @ V0 ) ) ).

thf(logicdef_2E_3D_3D_3E,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
    <=> ( V0
       => V1 ) ) ).

thf(logicdef_2E_3D,axiom,
    ! [A_27a: $tType,V0: A_27a,V1: A_27a] :
      ( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
    <=> ( V0 = V1 ) ) ).

thf(quantdef_2E_21,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_21 @ A_27a @ V0f )
    <=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(quantdef_2E_3F,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_3F @ A_27a @ V0f )
    <=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(thm_2Earithmetic_2EONE,axiom,
    ( ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) )
    = ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) ).

thf(thm_2Earithmetic_2ETWO,axiom,
    ( ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) )
    = ( c_2Enum_2ESUC @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).

thf(thm_2Earithmetic_2EADD__0,axiom,
    ! [V0m: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2E_2B @ V0m @ c_2Enum_2E0 )
      = V0m ) ).

thf(thm_2Earithmetic_2ELESS__MONO__EQ,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Eprim__rec_2E_3C @ ( c_2Enum_2ESUC @ V0m ) @ ( c_2Enum_2ESUC @ V1n ) )
      = ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).

thf(thm_2Earithmetic_2EADD1,axiom,
    ! [V0m: tyop_2Enum_2Enum] :
      ( ( c_2Enum_2ESUC @ V0m )
      = ( c_2Earithmetic_2E_2B @ V0m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).

thf(thm_2Earithmetic_2EMULT__COMM,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2E_2A @ V0m @ V1n )
      = ( c_2Earithmetic_2E_2A @ V1n @ V0m ) ) ).

thf(thm_2Earithmetic_2EEVEN__ODD,axiom,
    ! [V0n: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2EEVEN @ V0n )
    <=> ( (~) @ ( c_2Earithmetic_2EODD @ V0n ) ) ) ).

thf(thm_2Earithmetic_2EODD__EVEN,axiom,
    ! [V0n: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2EODD @ V0n )
    <=> ( (~) @ ( c_2Earithmetic_2EEVEN @ V0n ) ) ) ).

thf(thm_2Earithmetic_2EEVEN__EXISTS,axiom,
    ! [V0n: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2EEVEN @ V0n )
    <=> ? [V1m: tyop_2Enum_2Enum] :
          ( V0n
          = ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) ) ) ).

thf(thm_2Earithmetic_2EODD__EXISTS,axiom,
    ! [V0n: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2EODD @ V0n )
    <=> ? [V1m: tyop_2Enum_2Enum] :
          ( V0n
          = ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) ) ) ) ).

thf(thm_2Earithmetic_2EMOD__UNIQUE,axiom,
    ! [V0n: tyop_2Enum_2Enum,V1k: tyop_2Enum_2Enum,V2r: tyop_2Enum_2Enum] :
      ( ? [V3q: tyop_2Enum_2Enum] :
          ( ( V1k
            = ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V3q @ V0n ) @ V2r ) )
          & ( c_2Eprim__rec_2E_3C @ V2r @ V0n ) )
     => ( ( c_2Earithmetic_2EMOD @ V1k @ V0n )
        = V2r ) ) ).

thf(thm_2Ebool_2EBOOL__CASES__AX,axiom,
    ! [V0t: $o] :
      ( ( V0t = c_2Ebool_2ET )
      | ( V0t = c_2Ebool_2EF ) ) ).

thf(thm_2Ebool_2ETRUTH,axiom,
    c_2Ebool_2ET ).

thf(thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
    ! [V0t1: $o,V1t2: $o] :
      ( ( V0t1
       => V1t2 )
     => ( ( V1t2
         => V0t1 )
       => ( V0t1 = V1t2 ) ) ) ).

thf(thm_2Ebool_2EFALSITY,axiom,
    ! [V0t: $o] :
      ( c_2Ebool_2EF
     => V0t ) ).

thf(thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
    ! [V0t: $o] :
      ( V0t
      | ( (~) @ V0t ) ) ).

thf(thm_2Ebool_2EIMP__F,axiom,
    ! [V0t: $o] :
      ( ( V0t
       => c_2Ebool_2EF )
     => ( (~) @ V0t ) ) ).

thf(thm_2Ebool_2EF__IMP,axiom,
    ! [V0t: $o] :
      ( ( (~) @ V0t )
     => ( V0t
       => c_2Ebool_2EF ) ) ).

thf(thm_2Ebool_2EAND__CLAUSES,axiom,
    ! [V0t: $o] :
      ( ( ( c_2Ebool_2ET
          & V0t )
      <=> V0t )
      & ( ( V0t
          & c_2Ebool_2ET )
      <=> V0t )
      & ( ( c_2Ebool_2EF
          & V0t )
      <=> c_2Ebool_2EF )
      & ( ( V0t
          & c_2Ebool_2EF )
      <=> c_2Ebool_2EF )
      & ( ( V0t
          & V0t )
      <=> V0t ) ) ).

thf(thm_2Ebool_2EOR__CLAUSES,axiom,
    ! [V0t: $o] :
      ( ( ( c_2Ebool_2ET
          | V0t )
      <=> c_2Ebool_2ET )
      & ( ( V0t
          | c_2Ebool_2ET )
      <=> c_2Ebool_2ET )
      & ( ( c_2Ebool_2EF
          | V0t )
      <=> V0t )
      & ( ( V0t
          | c_2Ebool_2EF )
      <=> V0t )
      & ( ( V0t
          | V0t )
      <=> V0t ) ) ).

thf(thm_2Ebool_2EIMP__CLAUSES,axiom,
    ! [V0t: $o] :
      ( ( ( c_2Ebool_2ET
         => V0t )
      <=> V0t )
      & ( ( V0t
         => c_2Ebool_2ET )
      <=> c_2Ebool_2ET )
      & ( ( c_2Ebool_2EF
         => V0t )
      <=> c_2Ebool_2ET )
      & ( ( V0t
         => V0t )
      <=> c_2Ebool_2ET )
      & ( ( V0t
         => c_2Ebool_2EF )
      <=> ( (~) @ V0t ) ) ) ).

thf(thm_2Ebool_2ENOT__CLAUSES,axiom,
    ( ! [V0t: $o] :
        ( ( (~) @ ( (~) @ V0t ) )
      <=> V0t )
    & ( ( (~) @ c_2Ebool_2ET )
    <=> c_2Ebool_2EF )
    & ( ( (~) @ c_2Ebool_2EF )
    <=> c_2Ebool_2ET ) ) ).

thf(thm_2Ebool_2EREFL__CLAUSE,axiom,
    ! [A_27a: $tType,V0x: A_27a] :
      ( ( V0x = V0x )
    <=> c_2Ebool_2ET ) ).

thf(thm_2Ebool_2EEQ__SYM__EQ,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
      ( ( V0x = V1y )
    <=> ( V1y = V0x ) ) ).

thf(thm_2Ebool_2EEQ__CLAUSES,axiom,
    ! [V0t: $o] :
      ( ( ( c_2Ebool_2ET = V0t )
      <=> V0t )
      & ( ( V0t = c_2Ebool_2ET )
      <=> V0t )
      & ( ( c_2Ebool_2EF = V0t )
      <=> ( (~) @ V0t ) )
      & ( ( V0t = c_2Ebool_2EF )
      <=> ( (~) @ V0t ) ) ) ).

thf(thm_2Ebool_2ECOND__CLAUSES,axiom,
    ! [A_27a: $tType,V0t1: A_27a,V1t2: A_27a] :
      ( ( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2ET @ V0t1 @ V1t2 )
        = V0t1 )
      & ( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2EF @ V0t1 @ V1t2 )
        = V1t2 ) ) ).

thf(thm_2Ebool_2ENOT__EXISTS__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o] :
      ( ( (~)
        @ ? [V1x: A_27a] : ( V0P @ V1x ) )
    <=> ! [V2x: A_27a] : ( (~) @ ( V0P @ V2x ) ) ) ).

thf(thm_2Ebool_2ELEFT__AND__FORALL__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
      ( ( ! [V2x: A_27a] : ( V0P @ V2x )
        & V1Q )
    <=> ! [V3x: A_27a] :
          ( ( V0P @ V3x )
          & V1Q ) ) ).

thf(thm_2Ebool_2EEXISTS__OR__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
      ( ? [V2x: A_27a] :
          ( ( V0P @ V2x )
          | ( V1Q @ V2x ) )
    <=> ( ? [V3x: A_27a] : ( V0P @ V3x )
        | ? [V4x: A_27a] : ( V1Q @ V4x ) ) ) ).

thf(thm_2Ebool_2ELEFT__OR__EXISTS__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
      ( ( ? [V2x: A_27a] : ( V0P @ V2x )
        | V1Q )
    <=> ? [V3x: A_27a] :
          ( ( V0P @ V3x )
          | V1Q ) ) ).

thf(thm_2Ebool_2ERIGHT__OR__EXISTS__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
      ( ( V0P
        | ? [V2x: A_27a] : ( V1Q @ V2x ) )
    <=> ? [V3x: A_27a] :
          ( V0P
          | ( V1Q @ V3x ) ) ) ).

thf(thm_2Ebool_2ELEFT__EXISTS__AND__THM,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
      ( ? [V2x: A_27a] :
          ( ( V0P @ V2x )
          & V1Q )
    <=> ( ? [V3x: A_27a] : ( V0P @ V3x )
        & V1Q ) ) ).

thf(thm_2Ebool_2ERIGHT__EXISTS__AND__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
      ( ? [V2x: A_27a] :
          ( V0P
          & ( V1Q @ V2x ) )
    <=> ( V0P
        & ? [V3x: A_27a] : ( V1Q @ V3x ) ) ) ).

thf(thm_2Ebool_2ERIGHT__FORALL__OR__THM,axiom,
    ! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
      ( ! [V2x: A_27a] :
          ( V0P
          | ( V1Q @ V2x ) )
    <=> ( V0P
        | ! [V3x: A_27a] : ( V1Q @ V3x ) ) ) ).

thf(thm_2Ebool_2EDISJ__ASSOC,axiom,
    ! [V0A: $o,V1B: $o,V2C: $o] :
      ( ( V0A
        | V1B
        | V2C )
    <=> ( V0A
        | V1B
        | V2C ) ) ).

thf(thm_2Ebool_2EDISJ__SYM,axiom,
    ! [V0A: $o,V1B: $o] :
      ( ( V0A
        | V1B )
    <=> ( V1B
        | V0A ) ) ).

thf(thm_2Ebool_2EDE__MORGAN__THM,axiom,
    ! [V0A: $o,V1B: $o] :
      ( ( ( (~)
          @ ( V0A
            & V1B ) )
      <=> ( ( (~) @ V0A )
          | ( (~) @ V1B ) ) )
      & ( ( (~)
          @ ( V0A
            | V1B ) )
      <=> ( ( (~) @ V0A )
          & ( (~) @ V1B ) ) ) ) ).

thf(thm_2Ebool_2ECOND__RATOR,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0b: $o,V1f: A_27a > A_27b,V2g: A_27a > A_27b,V3x: A_27a] :
      ( ( c_2Ebool_2ECOND @ ( A_27a > A_27b ) @ V0b @ V1f @ V2g @ V3x )
      = ( c_2Ebool_2ECOND @ A_27b @ V0b @ ( V1f @ V3x ) @ ( V2g @ V3x ) ) ) ).

thf(thm_2Ebool_2ECOND__RAND,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1b: $o,V2x: A_27a,V3y: A_27a] :
      ( ( V0f @ ( c_2Ebool_2ECOND @ A_27a @ V1b @ V2x @ V3y ) )
      = ( c_2Ebool_2ECOND @ A_27b @ V1b @ ( V0f @ V2x ) @ ( V0f @ V3y ) ) ) ).

thf(thm_2Ebool_2ECOND__EXPAND,axiom,
    ! [V0b: $o,V1t1: $o,V2t2: $o] :
      ( ( c_2Ebool_2ECOND @ $o @ V0b @ V1t1 @ V2t2 )
    <=> ( ( ( (~) @ V0b )
          | V1t1 )
        & ( V0b
          | V2t2 ) ) ) ).

thf(thm_2Ebool_2ESKOLEM__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
      ( ! [V1x: A_27a] :
        ? [V2y: A_27b] : ( V0P @ V1x @ V2y )
    <=> ? [V3f: A_27a > A_27b] :
        ! [V4x: A_27a] : ( V0P @ V4x @ ( V3f @ V4x ) ) ) ).

thf(thm_2Eprim__rec_2ELESS__0,axiom,
    ! [V0n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Enum_2ESUC @ V0n ) ) ).

thf(thm_2Esat_2ENOT__NOT,axiom,
    ! [V0t: $o] :
      ( ( (~) @ ( (~) @ V0t ) )
    <=> V0t ) ).

thf(thm_2Esat_2EAND__INV__IMP,axiom,
    ! [V0A: $o] :
      ( V0A
     => ( ( (~) @ V0A )
       => c_2Ebool_2EF ) ) ).

thf(thm_2Esat_2EOR__DUAL2,axiom,
    ! [V0B: $o,V1A: $o] :
      ( ( ( (~)
          @ ( V1A
            | V0B ) )
       => c_2Ebool_2EF )
    <=> ( ( V1A
         => c_2Ebool_2EF )
       => ( ( (~) @ V0B )
         => c_2Ebool_2EF ) ) ) ).

thf(thm_2Esat_2EOR__DUAL3,axiom,
    ! [V0B: $o,V1A: $o] :
      ( ( ( (~)
          @ ( ( (~) @ V1A )
            | V0B ) )
       => c_2Ebool_2EF )
    <=> ( V1A
       => ( ( (~) @ V0B )
         => c_2Ebool_2EF ) ) ) ).

thf(thm_2Esat_2EAND__INV2,axiom,
    ! [V0A: $o] :
      ( ( ( (~) @ V0A )
       => c_2Ebool_2EF )
     => ( ( V0A
         => c_2Ebool_2EF )
       => c_2Ebool_2EF ) ) ).

thf(thm_2Esat_2Edc__eq,axiom,
    ! [V0r: $o,V1q: $o,V2p: $o] :
      ( ( V2p
      <=> ( V1q = V0r ) )
    <=> ( ( V2p
          | V1q
          | V0r )
        & ( V2p
          | ( (~) @ V0r )
          | ( (~) @ V1q ) )
        & ( V1q
          | ( (~) @ V0r )
          | ( (~) @ V2p ) )
        & ( V0r
          | ( (~) @ V1q )
          | ( (~) @ V2p ) ) ) ) ).

thf(thm_2Esat_2Edc__conj,axiom,
    ! [V0r: $o,V1q: $o,V2p: $o] :
      ( ( V2p
      <=> ( V1q
          & V0r ) )
    <=> ( ( V2p
          | ( (~) @ V1q )
          | ( (~) @ V0r ) )
        & ( V1q
          | ( (~) @ V2p ) )
        & ( V0r
          | ( (~) @ V2p ) ) ) ) ).

thf(thm_2Esat_2Edc__disj,axiom,
    ! [V0r: $o,V1q: $o,V2p: $o] :
      ( ( V2p
      <=> ( V1q
          | V0r ) )
    <=> ( ( V2p
          | ( (~) @ V1q ) )
        & ( V2p
          | ( (~) @ V0r ) )
        & ( V1q
          | V0r
          | ( (~) @ V2p ) ) ) ) ).

thf(thm_2Esat_2Edc__imp,axiom,
    ! [V0r: $o,V1q: $o,V2p: $o] :
      ( ( V2p
      <=> ( V1q
         => V0r ) )
    <=> ( ( V2p
          | V1q )
        & ( V2p
          | ( (~) @ V0r ) )
        & ( ( (~) @ V1q )
          | V0r
          | ( (~) @ V2p ) ) ) ) ).

thf(thm_2Esat_2Edc__neg,axiom,
    ! [V0q: $o,V1p: $o] :
      ( ( V1p
      <=> ( (~) @ V0q ) )
    <=> ( ( V1p
          | V0q )
        & ( ( (~) @ V0q )
          | ( (~) @ V1p ) ) ) ) ).

thf(thm_2Esat_2Epth__ni1,axiom,
    ! [V0q: $o,V1p: $o] :
      ( ( (~)
        @ ( V1p
         => V0q ) )
     => V1p ) ).

thf(thm_2Esat_2Epth__ni2,axiom,
    ! [V0q: $o,V1p: $o] :
      ( ( (~)
        @ ( V1p
         => V0q ) )
     => ( (~) @ V0q ) ) ).

thf(thm_2Esat_2Epth__no1,axiom,
    ! [V0q: $o,V1p: $o] :
      ( ( (~)
        @ ( V1p
          | V0q ) )
     => ( (~) @ V1p ) ) ).

thf(thm_2Esat_2Epth__no2,axiom,
    ! [V0q: $o,V1p: $o] :
      ( ( (~)
        @ ( V1p
          | V0q ) )
     => ( (~) @ V0q ) ) ).

thf(thm_2Esat_2Epth__nn,axiom,
    ! [V0p: $o] :
      ( ( (~) @ ( (~) @ V0p ) )
     => V0p ) ).

thf(thm_2Earithmetic_2EMOD__2,conjecture,
    ! [V0n: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2EMOD @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
      = ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2EEVEN @ V0n ) @ c_2Enum_2E0 @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).

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