ITP001 Axioms: ITP099^7.ax


%------------------------------------------------------------------------------
% File     : ITP099^7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy 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    : hreal.ax [Gau19]
%          : HL4099^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  107 (  38 unt;  28 typ;   0 def)
%            Number of atoms       :  231 (  48 equ;  15 cnn)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives :  598 (  15   ~;   7   |;  24   &; 514   @)
%                                         (  15 <=>;  23  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   14 (   6 avg; 514 nst)
%            Number of types       :    4 (   3 usr)
%            Number of type conns  :   55 (  55   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   27 (  25 usr;   3 con; 0-4 aty)
%            Number of variables   :  218 (  27   ^ 162   !;  24   ?; 218   :)
%                                         (   5  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Ehrat_2Ehrat,type,
    tyop_2Ehrat_2Ehrat: $tType ).

thf(tyop_2Ehreal_2Ehreal,type,
    tyop_2Ehreal_2Ehreal: $tType ).

thf(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

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

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

thf(c_2Ebool_2E_2F_5C,type,
    c_2Ebool_2E_2F_5C: $o > $o > $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_2Ebool_2ETYPE__DEFINITION,type,
    c_2Ebool_2ETYPE__DEFINITION: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).

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

thf(c_2Ehreal_2Ecut,type,
    c_2Ehreal_2Ecut: tyop_2Ehreal_2Ehreal > tyop_2Ehrat_2Ehrat > $o ).

thf(c_2Ehreal_2Ecut__of__hrat,type,
    c_2Ehreal_2Ecut__of__hrat: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat > $o ).

thf(c_2Ehrat_2Ehrat__1,type,
    c_2Ehrat_2Ehrat__1: tyop_2Ehrat_2Ehrat ).

thf(c_2Ehrat_2Ehrat__add,type,
    c_2Ehrat_2Ehrat__add: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat ).

thf(c_2Ehrat_2Ehrat__inv,type,
    c_2Ehrat_2Ehrat__inv: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat ).

thf(c_2Ehreal_2Ehrat__lt,type,
    c_2Ehreal_2Ehrat__lt: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat > $o ).

thf(c_2Ehrat_2Ehrat__mul,type,
    c_2Ehrat_2Ehrat__mul: tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat > tyop_2Ehrat_2Ehrat ).

thf(c_2Ehreal_2Ehreal,type,
    c_2Ehreal_2Ehreal: ( tyop_2Ehrat_2Ehrat > $o ) > tyop_2Ehreal_2Ehreal ).

thf(c_2Ehreal_2Ehreal__1,type,
    c_2Ehreal_2Ehreal__1: tyop_2Ehreal_2Ehreal ).

thf(c_2Ehreal_2Ehreal__add,type,
    c_2Ehreal_2Ehreal__add: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).

thf(c_2Ehreal_2Ehreal__inv,type,
    c_2Ehreal_2Ehreal__inv: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).

thf(c_2Ehreal_2Ehreal__lt,type,
    c_2Ehreal_2Ehreal__lt: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal > $o ).

thf(c_2Ehreal_2Ehreal__mul,type,
    c_2Ehreal_2Ehreal__mul: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).

thf(c_2Ehreal_2Ehreal__sub,type,
    c_2Ehreal_2Ehreal__sub: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).

thf(c_2Ehreal_2Ehreal__sup,type,
    c_2Ehreal_2Ehreal__sup: ( tyop_2Ehreal_2Ehreal > $o ) > tyop_2Ehreal_2Ehreal ).

thf(c_2Ehreal_2Eisacut,type,
    c_2Ehreal_2Eisacut: ( tyop_2Ehrat_2Ehrat > $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_2Ehreal_2Ehrat__lt,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
    <=> ? [V2d: tyop_2Ehrat_2Ehrat] :
          ( V1y
          = ( c_2Ehrat_2Ehrat__add @ V0x @ V2d ) ) ) ).

thf(thm_2Ehreal_2Eisacut,axiom,
    ! [V0C: tyop_2Ehrat_2Ehrat > $o] :
      ( ( c_2Ehreal_2Eisacut @ V0C )
    <=> ( ? [V1x: tyop_2Ehrat_2Ehrat] : ( V0C @ V1x )
        & ? [V2x: tyop_2Ehrat_2Ehrat] : ( (~) @ ( V0C @ V2x ) )
        & ! [V3x: tyop_2Ehrat_2Ehrat,V4y: tyop_2Ehrat_2Ehrat] :
            ( ( ( V0C @ V3x )
              & ( c_2Ehreal_2Ehrat__lt @ V4y @ V3x ) )
           => ( V0C @ V4y ) )
        & ! [V5x: tyop_2Ehrat_2Ehrat] :
            ( ( V0C @ V5x )
           => ? [V6y: tyop_2Ehrat_2Ehrat] :
                ( ( V0C @ V6y )
                & ( c_2Ehreal_2Ehrat__lt @ V5x @ V6y ) ) ) ) ) ).

thf(thm_2Ehreal_2Ecut__of__hrat,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ecut__of__hrat @ V0x )
      = ( ^ [V1y: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt @ V1y @ V0x ) ) ) ).

thf(thm_2Ehreal_2Ehreal__TY__DEF,axiom,
    ? [V0rep: tyop_2Ehreal_2Ehreal > tyop_2Ehrat_2Ehrat > $o] : ( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Ehrat_2Ehrat > $o ) @ tyop_2Ehreal_2Ehreal @ c_2Ehreal_2Eisacut @ V0rep ) ).

thf(thm_2Ehreal_2Ehreal__tybij,axiom,
    ( ! [V0a: tyop_2Ehreal_2Ehreal] :
        ( ( c_2Ehreal_2Ehreal @ ( c_2Ehreal_2Ecut @ V0a ) )
        = V0a )
    & ! [V1r: tyop_2Ehrat_2Ehrat > $o] :
        ( ( c_2Ehreal_2Eisacut @ V1r )
      <=> ( ( c_2Ehreal_2Ecut @ ( c_2Ehreal_2Ehreal @ V1r ) )
          = V1r ) ) ) ).

thf(thm_2Ehreal_2Ehreal__1,axiom,
    ( c_2Ehreal_2Ehreal__1
    = ( c_2Ehreal_2Ehreal @ ( c_2Ehreal_2Ecut__of__hrat @ c_2Ehrat_2Ehrat__1 ) ) ) ).

thf(thm_2Ehreal_2Ehreal__add,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__add @ V0X @ V1Y )
      = ( c_2Ehreal_2Ehreal
        @ ^ [V2w: tyop_2Ehrat_2Ehrat] :
            ( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
            @ ^ [V3x: tyop_2Ehrat_2Ehrat] :
                ( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
                @ ^ [V4y: tyop_2Ehrat_2Ehrat] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ tyop_2Ehrat_2Ehrat @ V2w @ ( c_2Ehrat_2Ehrat__add @ V3x @ V4y ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) @ ( c_2Ehreal_2Ecut @ V1Y @ V4y ) ) ) ) ) ) ) ).

thf(thm_2Ehreal_2Ehreal__mul,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__mul @ V0X @ V1Y )
      = ( c_2Ehreal_2Ehreal
        @ ^ [V2w: tyop_2Ehrat_2Ehrat] :
            ( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
            @ ^ [V3x: tyop_2Ehrat_2Ehrat] :
                ( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
                @ ^ [V4y: tyop_2Ehrat_2Ehrat] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ tyop_2Ehrat_2Ehrat @ V2w @ ( c_2Ehrat_2Ehrat__mul @ V3x @ V4y ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) @ ( c_2Ehreal_2Ecut @ V1Y @ V4y ) ) ) ) ) ) ) ).

thf(thm_2Ehreal_2Ehreal__inv,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__inv @ V0X )
      = ( c_2Ehreal_2Ehreal
        @ ^ [V1w: tyop_2Ehrat_2Ehrat] :
            ( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
            @ ^ [V2d: tyop_2Ehrat_2Ehrat] :
                ( c_2Ebool_2E_2F_5C @ ( c_2Ehreal_2Ehrat__lt @ V2d @ c_2Ehrat_2Ehrat__1 )
                @ ( c_2Ebool_2E_21 @ tyop_2Ehrat_2Ehrat
                  @ ^ [V3x: tyop_2Ehrat_2Ehrat] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) @ ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V1w @ V3x ) @ V2d ) ) ) ) ) ) ) ).

thf(thm_2Ehreal_2Ehreal__sup,axiom,
    ! [V0P: tyop_2Ehreal_2Ehreal > $o] :
      ( ( c_2Ehreal_2Ehreal__sup @ V0P )
      = ( c_2Ehreal_2Ehreal
        @ ^ [V1w: tyop_2Ehrat_2Ehrat] :
            ( c_2Ebool_2E_3F @ tyop_2Ehreal_2Ehreal
            @ ^ [V2X: tyop_2Ehreal_2Ehreal] : ( c_2Ebool_2E_2F_5C @ ( V0P @ V2X ) @ ( c_2Ehreal_2Ecut @ V2X @ V1w ) ) ) ) ) ).

thf(thm_2Ehreal_2Ehreal__lt,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__lt @ V0X @ V1Y )
    <=> ( ( (~) @ ( V0X = V1Y ) )
        & ! [V2x: tyop_2Ehrat_2Ehrat] :
            ( ( c_2Ehreal_2Ecut @ V0X @ V2x )
           => ( c_2Ehreal_2Ecut @ V1Y @ V2x ) ) ) ) ).

thf(thm_2Ehreal_2Ehreal__sub,axiom,
    ! [V0Y: tyop_2Ehreal_2Ehreal,V1X: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__sub @ V0Y @ V1X )
      = ( c_2Ehreal_2Ehreal
        @ ^ [V2w: tyop_2Ehrat_2Ehrat] :
            ( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
            @ ^ [V3x: tyop_2Ehrat_2Ehrat] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_7E @ ( c_2Ehreal_2Ecut @ V1X @ V3x ) ) @ ( c_2Ehreal_2Ecut @ V0Y @ ( c_2Ehrat_2Ehrat__add @ V3x @ V2w ) ) ) ) ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__REFL,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat] : ( (~) @ ( c_2Ehreal_2Ehrat__lt @ V0x @ V0x ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__TRANS,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
      ( ( ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
        & ( c_2Ehreal_2Ehrat__lt @ V1y @ V2z ) )
     => ( c_2Ehreal_2Ehrat__lt @ V0x @ V2z ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__ANTISYM,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( (~)
      @ ( ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
        & ( c_2Ehreal_2Ehrat__lt @ V1y @ V0x ) ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__TOTAL,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( ( V0x = V1y )
      | ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
      | ( c_2Ehreal_2Ehrat__lt @ V1y @ V0x ) ) ).

thf(thm_2Ehreal_2EHRAT__MUL__RID,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehrat_2Ehrat__mul @ V0x @ c_2Ehrat_2Ehrat__1 )
      = V0x ) ).

thf(thm_2Ehreal_2EHRAT__MUL__RINV,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehrat_2Ehrat__mul @ V0x @ ( c_2Ehrat_2Ehrat__inv @ V0x ) )
      = c_2Ehrat_2Ehrat__1 ) ).

thf(thm_2Ehreal_2EHRAT__RDISTRIB,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehrat_2Ehrat__mul @ ( c_2Ehrat_2Ehrat__add @ V0x @ V1y ) @ V2z )
      = ( c_2Ehrat_2Ehrat__add @ ( c_2Ehrat_2Ehrat__mul @ V0x @ V2z ) @ ( c_2Ehrat_2Ehrat__mul @ V1y @ V2z ) ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__ADDL,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt @ V0x @ ( c_2Ehrat_2Ehrat__add @ V0x @ V1y ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__ADDR,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt @ V1y @ ( c_2Ehrat_2Ehrat__add @ V0x @ V1y ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__GT,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
     => ( (~) @ ( c_2Ehreal_2Ehrat__lt @ V1y @ V0x ) ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__NE,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
     => ( (~) @ ( V0x = V1y ) ) ) ).

thf(thm_2Ehreal_2EHRAT__EQ__LADD,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
      ( ( ( c_2Ehrat_2Ehrat__add @ V0x @ V1y )
        = ( c_2Ehrat_2Ehrat__add @ V0x @ V2z ) )
    <=> ( V1y = V2z ) ) ).

thf(thm_2Ehreal_2EHRAT__EQ__LMUL,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
      ( ( ( c_2Ehrat_2Ehrat__mul @ V0x @ V1y )
        = ( c_2Ehrat_2Ehrat__mul @ V0x @ V2z ) )
    <=> ( V1y = V2z ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__ADD2,axiom,
    ! [V0u: tyop_2Ehrat_2Ehrat,V1v: tyop_2Ehrat_2Ehrat,V2x: tyop_2Ehrat_2Ehrat,V3y: tyop_2Ehrat_2Ehrat] :
      ( ( ( c_2Ehreal_2Ehrat__lt @ V0u @ V2x )
        & ( c_2Ehreal_2Ehrat__lt @ V1v @ V3y ) )
     => ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__add @ V0u @ V1v ) @ ( c_2Ehrat_2Ehrat__add @ V2x @ V3y ) ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__LADD,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__add @ V2z @ V0x ) @ ( c_2Ehrat_2Ehrat__add @ V2z @ V1y ) )
      = ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__RADD,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__add @ V0x @ V2z ) @ ( c_2Ehrat_2Ehrat__add @ V1y @ V2z ) )
      = ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__MUL2,axiom,
    ! [V0u: tyop_2Ehrat_2Ehrat,V1v: tyop_2Ehrat_2Ehrat,V2x: tyop_2Ehrat_2Ehrat,V3y: tyop_2Ehrat_2Ehrat] :
      ( ( ( c_2Ehreal_2Ehrat__lt @ V0u @ V2x )
        & ( c_2Ehreal_2Ehrat__lt @ V1v @ V3y ) )
     => ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V0u @ V1v ) @ ( c_2Ehrat_2Ehrat__mul @ V2x @ V3y ) ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__LMUL,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V2z @ V0x ) @ ( c_2Ehrat_2Ehrat__mul @ V2z @ V1y ) )
      = ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__RMUL,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat,V2z: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V0x @ V2z ) @ ( c_2Ehrat_2Ehrat__mul @ V1y @ V2z ) )
      = ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__LMUL1,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V0x @ V1y ) @ V1y )
      = ( c_2Ehreal_2Ehrat__lt @ V0x @ c_2Ehrat_2Ehrat__1 ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__RMUL1,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V0x @ V1y ) @ V0x )
      = ( c_2Ehreal_2Ehrat__lt @ V1y @ c_2Ehrat_2Ehrat__1 ) ) ).

thf(thm_2Ehreal_2EHRAT__GT__LMUL1,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ V1y @ ( c_2Ehrat_2Ehrat__mul @ V0x @ V1y ) )
      = ( c_2Ehreal_2Ehrat__lt @ c_2Ehrat_2Ehrat__1 @ V0x ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__L1,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ ( c_2Ehrat_2Ehrat__inv @ V0x ) @ V1y ) @ c_2Ehrat_2Ehrat__1 )
      = ( c_2Ehreal_2Ehrat__lt @ V1y @ V0x ) ) ).

thf(thm_2Ehreal_2EHRAT__LT__R1,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V0x @ ( c_2Ehrat_2Ehrat__inv @ V1y ) ) @ c_2Ehrat_2Ehrat__1 )
      = ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ) ).

thf(thm_2Ehreal_2EHRAT__GT__L1,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ c_2Ehrat_2Ehrat__1 @ ( c_2Ehrat_2Ehrat__mul @ ( c_2Ehrat_2Ehrat__inv @ V0x ) @ V1y ) )
      = ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ) ).

thf(thm_2Ehreal_2EHRAT__INV__MUL,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehrat_2Ehrat__inv @ ( c_2Ehrat_2Ehrat__mul @ V0x @ V1y ) )
      = ( c_2Ehrat_2Ehrat__mul @ ( c_2Ehrat_2Ehrat__inv @ V0x ) @ ( c_2Ehrat_2Ehrat__inv @ V1y ) ) ) ).

thf(thm_2Ehreal_2EHRAT__UP,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat] :
    ? [V1y: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y ) ).

thf(thm_2Ehreal_2EHRAT__DOWN,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat] :
    ? [V1y: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ehrat__lt @ V1y @ V0x ) ).

thf(thm_2Ehreal_2EHRAT__DOWN2,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
    ? [V2z: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ V2z @ V0x )
      & ( c_2Ehreal_2Ehrat__lt @ V2z @ V1y ) ) ).

thf(thm_2Ehreal_2EHRAT__MEAN,axiom,
    ! [V0x: tyop_2Ehrat_2Ehrat,V1y: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ V0x @ V1y )
     => ? [V2z: tyop_2Ehrat_2Ehrat] :
          ( ( c_2Ehreal_2Ehrat__lt @ V0x @ V2z )
          & ( c_2Ehreal_2Ehrat__lt @ V2z @ V1y ) ) ) ).

thf(thm_2Ehreal_2EISACUT__HRAT,axiom,
    ! [V0h: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Eisacut @ ( c_2Ehreal_2Ecut__of__hrat @ V0h ) ) ).

thf(thm_2Ehreal_2EEQUAL__CUTS,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( ( ( c_2Ehreal_2Ecut @ V0X )
        = ( c_2Ehreal_2Ecut @ V1Y ) )
     => ( V0X = V1Y ) ) ).

thf(thm_2Ehreal_2ECUT__ISACUT,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Eisacut @ ( c_2Ehreal_2Ecut @ V0X ) ) ).

thf(thm_2Ehreal_2ECUT__NONEMPTY,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal] :
    ? [V1x: tyop_2Ehrat_2Ehrat] : ( c_2Ehreal_2Ecut @ V0X @ V1x ) ).

thf(thm_2Ehreal_2ECUT__BOUNDED,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal] :
    ? [V1x: tyop_2Ehrat_2Ehrat] : ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ V1x ) ) ).

thf(thm_2Ehreal_2ECUT__DOWN,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1x: tyop_2Ehrat_2Ehrat,V2y: tyop_2Ehrat_2Ehrat] :
      ( ( ( c_2Ehreal_2Ecut @ V0X @ V1x )
        & ( c_2Ehreal_2Ehrat__lt @ V2y @ V1x ) )
     => ( c_2Ehreal_2Ecut @ V0X @ V2y ) ) ).

thf(thm_2Ehreal_2ECUT__UP,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1x: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ecut @ V0X @ V1x )
     => ? [V2y: tyop_2Ehrat_2Ehrat] :
          ( ( c_2Ehreal_2Ecut @ V0X @ V2y )
          & ( c_2Ehreal_2Ehrat__lt @ V1x @ V2y ) ) ) ).

thf(thm_2Ehreal_2ECUT__UBOUND,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1x: tyop_2Ehrat_2Ehrat,V2y: tyop_2Ehrat_2Ehrat] :
      ( ( ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ V1x ) )
        & ( c_2Ehreal_2Ehrat__lt @ V1x @ V2y ) )
     => ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ V2y ) ) ) ).

thf(thm_2Ehreal_2ECUT__STRADDLE,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1x: tyop_2Ehrat_2Ehrat,V2y: tyop_2Ehrat_2Ehrat] :
      ( ( ( c_2Ehreal_2Ecut @ V0X @ V1x )
        & ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ V2y ) ) )
     => ( c_2Ehreal_2Ehrat__lt @ V1x @ V2y ) ) ).

thf(thm_2Ehreal_2ECUT__NEARTOP__ADD,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1e: tyop_2Ehrat_2Ehrat] :
    ? [V2x: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ecut @ V0X @ V2x )
      & ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ ( c_2Ehrat_2Ehrat__add @ V2x @ V1e ) ) ) ) ).

thf(thm_2Ehreal_2ECUT__NEARTOP__MUL,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1u: tyop_2Ehrat_2Ehrat] :
      ( ( c_2Ehreal_2Ehrat__lt @ c_2Ehrat_2Ehrat__1 @ V1u )
     => ? [V2x: tyop_2Ehrat_2Ehrat] :
          ( ( c_2Ehreal_2Ecut @ V0X @ V2x )
          & ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ ( c_2Ehrat_2Ehrat__mul @ V1u @ V2x ) ) ) ) ) ).

thf(thm_2Ehreal_2EHREAL__INV__ISACUT,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal] :
      ( c_2Ehreal_2Eisacut
      @ ^ [V1w: tyop_2Ehrat_2Ehrat] :
          ( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
          @ ^ [V2d: tyop_2Ehrat_2Ehrat] :
              ( c_2Ebool_2E_2F_5C @ ( c_2Ehreal_2Ehrat__lt @ V2d @ c_2Ehrat_2Ehrat__1 )
              @ ( c_2Ebool_2E_21 @ tyop_2Ehrat_2Ehrat
                @ ^ [V3x: tyop_2Ehrat_2Ehrat] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) @ ( c_2Ehreal_2Ehrat__lt @ ( c_2Ehrat_2Ehrat__mul @ V1w @ V3x ) @ V2d ) ) ) ) ) ) ).

thf(thm_2Ehreal_2EHREAL__ADD__ISACUT,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( c_2Ehreal_2Eisacut
      @ ^ [V2w: tyop_2Ehrat_2Ehrat] :
          ( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
          @ ^ [V3x: tyop_2Ehrat_2Ehrat] :
              ( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
              @ ^ [V4y: tyop_2Ehrat_2Ehrat] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ tyop_2Ehrat_2Ehrat @ V2w @ ( c_2Ehrat_2Ehrat__add @ V3x @ V4y ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) @ ( c_2Ehreal_2Ecut @ V1Y @ V4y ) ) ) ) ) ) ).

thf(thm_2Ehreal_2EHREAL__MUL__ISACUT,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( c_2Ehreal_2Eisacut
      @ ^ [V2w: tyop_2Ehrat_2Ehrat] :
          ( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
          @ ^ [V3x: tyop_2Ehrat_2Ehrat] :
              ( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
              @ ^ [V4y: tyop_2Ehrat_2Ehrat] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ tyop_2Ehrat_2Ehrat @ V2w @ ( c_2Ehrat_2Ehrat__mul @ V3x @ V4y ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) @ ( c_2Ehreal_2Ecut @ V1Y @ V4y ) ) ) ) ) ) ).

thf(thm_2Ehreal_2EHREAL__ADD__SYM,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__add @ V0X @ V1Y )
      = ( c_2Ehreal_2Ehreal__add @ V1Y @ V0X ) ) ).

thf(thm_2Ehreal_2EHREAL__MUL__SYM,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__mul @ V0X @ V1Y )
      = ( c_2Ehreal_2Ehreal__mul @ V1Y @ V0X ) ) ).

thf(thm_2Ehreal_2EHREAL__ADD__ASSOC,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal,V2Z: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__add @ V0X @ ( c_2Ehreal_2Ehreal__add @ V1Y @ V2Z ) )
      = ( c_2Ehreal_2Ehreal__add @ ( c_2Ehreal_2Ehreal__add @ V0X @ V1Y ) @ V2Z ) ) ).

thf(thm_2Ehreal_2EHREAL__MUL__ASSOC,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal,V2Z: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__mul @ V0X @ ( c_2Ehreal_2Ehreal__mul @ V1Y @ V2Z ) )
      = ( c_2Ehreal_2Ehreal__mul @ ( c_2Ehreal_2Ehreal__mul @ V0X @ V1Y ) @ V2Z ) ) ).

thf(thm_2Ehreal_2EHREAL__LDISTRIB,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal,V2Z: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__mul @ V0X @ ( c_2Ehreal_2Ehreal__add @ V1Y @ V2Z ) )
      = ( c_2Ehreal_2Ehreal__add @ ( c_2Ehreal_2Ehreal__mul @ V0X @ V1Y ) @ ( c_2Ehreal_2Ehreal__mul @ V0X @ V2Z ) ) ) ).

thf(thm_2Ehreal_2EHREAL__MUL__LID,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__mul @ c_2Ehreal_2Ehreal__1 @ V0X )
      = V0X ) ).

thf(thm_2Ehreal_2EHREAL__MUL__LINV,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__mul @ ( c_2Ehreal_2Ehreal__inv @ V0X ) @ V0X )
      = c_2Ehreal_2Ehreal__1 ) ).

thf(thm_2Ehreal_2EHREAL__NOZERO,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( (~)
      @ ( ( c_2Ehreal_2Ehreal__add @ V0X @ V1Y )
        = V0X ) ) ).

thf(thm_2Ehreal_2EHREAL__LT__LEMMA,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__lt @ V0X @ V1Y )
     => ? [V2x: tyop_2Ehrat_2Ehrat] :
          ( ( (~) @ ( c_2Ehreal_2Ecut @ V0X @ V2x ) )
          & ( c_2Ehreal_2Ecut @ V1Y @ V2x ) ) ) ).

thf(thm_2Ehreal_2EHREAL__SUB__ISACUT,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__lt @ V0X @ V1Y )
     => ( c_2Ehreal_2Eisacut
        @ ^ [V2w: tyop_2Ehrat_2Ehrat] :
            ( c_2Ebool_2E_3F @ tyop_2Ehrat_2Ehrat
            @ ^ [V3x: tyop_2Ehrat_2Ehrat] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_7E @ ( c_2Ehreal_2Ecut @ V0X @ V3x ) ) @ ( c_2Ehreal_2Ecut @ V1Y @ ( c_2Ehrat_2Ehrat__add @ V3x @ V2w ) ) ) ) ) ) ).

thf(thm_2Ehreal_2EHREAL__SUB__ADD,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__lt @ V0X @ V1Y )
     => ( ( c_2Ehreal_2Ehreal__add @ ( c_2Ehreal_2Ehreal__sub @ V1Y @ V0X ) @ V0X )
        = V1Y ) ) ).

thf(thm_2Ehreal_2EHREAL__LT__TOTAL,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( ( V0X = V1Y )
      | ( c_2Ehreal_2Ehreal__lt @ V0X @ V1Y )
      | ( c_2Ehreal_2Ehreal__lt @ V1Y @ V0X ) ) ).

thf(thm_2Ehreal_2EHREAL__LT,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( ( c_2Ehreal_2Ehreal__lt @ V0X @ V1Y )
    <=> ? [V2D: tyop_2Ehreal_2Ehreal] :
          ( V1Y
          = ( c_2Ehreal_2Ehreal__add @ V0X @ V2D ) ) ) ).

thf(thm_2Ehreal_2EHREAL__ADD__TOTAL,axiom,
    ! [V0X: tyop_2Ehreal_2Ehreal,V1Y: tyop_2Ehreal_2Ehreal] :
      ( ( V0X = V1Y )
      | ? [V2D: tyop_2Ehreal_2Ehreal] :
          ( V1Y
          = ( c_2Ehreal_2Ehreal__add @ V0X @ V2D ) )
      | ? [V3D: tyop_2Ehreal_2Ehreal] :
          ( V0X
          = ( c_2Ehreal_2Ehreal__add @ V1Y @ V3D ) ) ) ).

thf(thm_2Ehreal_2EHREAL__SUP__ISACUT,axiom,
    ! [V0P: tyop_2Ehreal_2Ehreal > $o] :
      ( ( ? [V1X: tyop_2Ehreal_2Ehreal] : ( V0P @ V1X )
        & ? [V2Y: tyop_2Ehreal_2Ehreal] :
          ! [V3X: tyop_2Ehreal_2Ehreal] :
            ( ( V0P @ V3X )
           => ( c_2Ehreal_2Ehreal__lt @ V3X @ V2Y ) ) )
     => ( c_2Ehreal_2Eisacut
        @ ^ [V4w: tyop_2Ehrat_2Ehrat] :
            ( c_2Ebool_2E_3F @ tyop_2Ehreal_2Ehreal
            @ ^ [V5X: tyop_2Ehreal_2Ehreal] : ( c_2Ebool_2E_2F_5C @ ( V0P @ V5X ) @ ( c_2Ehreal_2Ecut @ V5X @ V4w ) ) ) ) ) ).

thf(thm_2Ehreal_2EHREAL__SUP,axiom,
    ! [V0P: tyop_2Ehreal_2Ehreal > $o] :
      ( ( ? [V1X: tyop_2Ehreal_2Ehreal] : ( V0P @ V1X )
        & ? [V2Y: tyop_2Ehreal_2Ehreal] :
          ! [V3X: tyop_2Ehreal_2Ehreal] :
            ( ( V0P @ V3X )
           => ( c_2Ehreal_2Ehreal__lt @ V3X @ V2Y ) ) )
     => ! [V4Y: tyop_2Ehreal_2Ehreal] :
          ( ? [V5X: tyop_2Ehreal_2Ehreal] :
              ( ( V0P @ V5X )
              & ( c_2Ehreal_2Ehreal__lt @ V4Y @ V5X ) )
        <=> ( c_2Ehreal_2Ehreal__lt @ V4Y @ ( c_2Ehreal_2Ehreal__sup @ V0P ) ) ) ) ).

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