ITP001 Axioms: ITP045^7.ax


%------------------------------------------------------------------------------
% File     : ITP045^7 : 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    : bag.ax [Gau19]
%          : HL4045^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  348 (  73 unt;  81 typ;   0 def)
%            Number of atoms       :  893 ( 234 equ;  43 cnn)
%            Maximal formula atoms :   66 (   2 avg)
%            Number of connectives : 3999 (  43   ~;  19   |; 177   &;3481   @)
%                                         (  83 <=>; 196  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   38 (   8 avg;3481 nst)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :  956 ( 956   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   81 (  79 usr;   4 con; 0-6 aty)
%            Number of variables   : 1166 (  20   ^1046   !;  32   ?;1166   :)
%                                         (  68  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
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(tyop_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: $tType > $tType > $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_2Epair_2E_2C,type,
    c_2Epair_2E_2C: 
      !>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).

thf(c_2Earithmetic_2E_2D,type,
    c_2Earithmetic_2E_2D: 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_2Earithmetic_2E_3C_3D,type,
    c_2Earithmetic_2E_3C_3D: 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_2Earithmetic_2E_3E,type,
    c_2Earithmetic_2E_3E: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).

thf(c_2Earithmetic_2E_3E_3D,type,
    c_2Earithmetic_2E_3E_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).

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

thf(c_2Ebag_2EBAG__ALL__DISTINCT,type,
    c_2Ebag_2EBAG__ALL__DISTINCT: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > $o ) ).

thf(c_2Ebag_2EBAG__CARD,type,
    c_2Ebag_2EBAG__CARD: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ) ).

thf(c_2Ebag_2EBAG__CARD__RELn,type,
    c_2Ebag_2EBAG__CARD__RELn: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > tyop_2Enum_2Enum > $o ) ).

thf(c_2Ebag_2EBAG__CHOICE,type,
    c_2Ebag_2EBAG__CHOICE: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > A_27a ) ).

thf(c_2Ebag_2EBAG__DELETE,type,
    c_2Ebag_2EBAG__DELETE: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > A_27a > ( A_27a > tyop_2Enum_2Enum ) > $o ) ).

thf(c_2Ebag_2EBAG__DIFF,type,
    c_2Ebag_2EBAG__DIFF: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).

thf(c_2Ebag_2EBAG__DISJOINT,type,
    c_2Ebag_2EBAG__DISJOINT: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > tyop_2Enum_2Enum ) > $o ) ).

thf(c_2Ebag_2EBAG__EVERY,type,
    c_2Ebag_2EBAG__EVERY: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > tyop_2Enum_2Enum ) > $o ) ).

thf(c_2Ebag_2EBAG__FILTER,type,
    c_2Ebag_2EBAG__FILTER: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).

thf(c_2Ebag_2EBAG__GEN__PROD,type,
    c_2Ebag_2EBAG__GEN__PROD: ( tyop_2Enum_2Enum > tyop_2Enum_2Enum ) > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Ebag_2EBAG__GEN__SUM,type,
    c_2Ebag_2EBAG__GEN__SUM: ( tyop_2Enum_2Enum > tyop_2Enum_2Enum ) > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Ebag_2EBAG__IMAGE,type,
    c_2Ebag_2EBAG__IMAGE: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a ) > ( A_27b > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).

thf(c_2Ebag_2EBAG__IN,type,
    c_2Ebag_2EBAG__IN: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > tyop_2Enum_2Enum ) > $o ) ).

thf(c_2Ebag_2EBAG__INN,type,
    c_2Ebag_2EBAG__INN: 
      !>[A_27a: $tType] : ( A_27a > tyop_2Enum_2Enum > ( A_27a > tyop_2Enum_2Enum ) > $o ) ).

thf(c_2Ebag_2EBAG__INSERT,type,
    c_2Ebag_2EBAG__INSERT: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).

thf(c_2Ebag_2EBAG__INTER,type,
    c_2Ebag_2EBAG__INTER: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).

thf(c_2Ebag_2EBAG__MERGE,type,
    c_2Ebag_2EBAG__MERGE: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).

thf(c_2Ebag_2EBAG__OF__SET,type,
    c_2Ebag_2EBAG__OF__SET: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > tyop_2Enum_2Enum ) ).

thf(c_2Ebag_2EBAG__REST,type,
    c_2Ebag_2EBAG__REST: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).

thf(c_2Ebag_2EBAG__UNION,type,
    c_2Ebag_2EBAG__UNION: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > tyop_2Enum_2Enum ) > A_27a > tyop_2Enum_2Enum ) ).

thf(c_2Ebag_2EBIG__BAG__UNION,type,
    c_2Ebag_2EBIG__BAG__UNION: 
      !>[A_27a: $tType] : ( ( ( A_27a > tyop_2Enum_2Enum ) > $o ) > A_27a > tyop_2Enum_2Enum ) ).

thf(c_2Earithmetic_2EBIT1,type,
    c_2Earithmetic_2EBIT1: 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_2Epred__set_2EDELETE,type,
    c_2Epred__set_2EDELETE: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).

thf(c_2Epred__set_2EDIFF,type,
    c_2Epred__set_2EDIFF: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).

thf(c_2Epred__set_2EDISJOINT,type,
    c_2Epred__set_2EDISJOINT: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).

thf(c_2Ebag_2EEL__BAG,type,
    c_2Ebag_2EEL__BAG: 
      !>[A_27a: $tType] : ( A_27a > A_27a > tyop_2Enum_2Enum ) ).

thf(c_2Epred__set_2EEMPTY,type,
    c_2Epred__set_2EEMPTY: 
      !>[A_27a: $tType] : ( A_27a > $o ) ).

thf(c_2Ebag_2EEMPTY__BAG,type,
    c_2Ebag_2EEMPTY__BAG: 
      !>[A_27a: $tType] : ( A_27a > tyop_2Enum_2Enum ) ).

thf(c_2Epred__set_2EFINITE,type,
    c_2Epred__set_2EFINITE: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Ebag_2EFINITE__BAG,type,
    c_2Ebag_2EFINITE__BAG: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > $o ) ).

thf(c_2Epred__set_2EGSPEC,type,
    c_2Epred__set_2EGSPEC: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).

thf(c_2Ecombin_2EI,type,
    c_2Ecombin_2EI: 
      !>[A_27a: $tType] : ( A_27a > A_27a ) ).

thf(c_2Epred__set_2EIMAGE,type,
    c_2Epred__set_2EIMAGE: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).

thf(c_2Ebool_2EIN,type,
    c_2Ebool_2EIN: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).

thf(c_2Epred__set_2EINSERT,type,
    c_2Epred__set_2EINSERT: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).

thf(c_2Epred__set_2EINTER,type,
    c_2Epred__set_2EINTER: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).

thf(c_2Ebag_2EITBAG,type,
    c_2Ebag_2EITBAG: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > A_27b ) > ( A_27a > tyop_2Enum_2Enum ) > A_27b > A_27b ) ).

thf(c_2Epred__set_2EITSET,type,
    c_2Epred__set_2EITSET: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > A_27b ) > ( A_27a > $o ) > A_27b > A_27b ) ).

thf(c_2Ecombin_2EK,type,
    c_2Ecombin_2EK: 
      !>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > A_27a ) ).

thf(c_2Ebool_2ELET,type,
    c_2Ebool_2ELET: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).

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

thf(c_2Ebag_2EPSUB__BAG,type,
    c_2Ebag_2EPSUB__BAG: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > tyop_2Enum_2Enum ) > $o ) ).

thf(c_2Ebag_2ESET__OF__BAG,type,
    c_2Ebag_2ESET__OF__BAG: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > A_27a > $o ) ).

thf(c_2Ebag_2ESING__BAG,type,
    c_2Ebag_2ESING__BAG: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > $o ) ).

thf(c_2Epred__set_2ESUBSET,type,
    c_2Epred__set_2ESUBSET: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).

thf(c_2Ebag_2ESUB__BAG,type,
    c_2Ebag_2ESUB__BAG: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > tyop_2Enum_2Enum ) > $o ) ).

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

thf(c_2Epred__set_2ESUM__IMAGE,type,
    c_2Epred__set_2ESUM__IMAGE: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > $o ) > tyop_2Enum_2Enum ) ).

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

thf(c_2Erelation_2ETC,type,
    c_2Erelation_2ETC: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).

thf(c_2Epred__set_2EUNION,type,
    c_2Epred__set_2EUNION: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).

thf(c_2Erelation_2EWF,type,
    c_2Erelation_2EWF: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).

thf(c_2Erelation_2EWFP,type,
    c_2Erelation_2EWFP: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > $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_2Ebag_2Ebag__size,type,
    c_2Ebag_2Ebag__size: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ) ).

thf(c_2Edivides_2Edivides,type,
    c_2Edivides_2Edivides: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).

thf(c_2Ebag_2Edominates,type,
    c_2Ebag_2Edominates: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).

thf(c_2Ebag_2Emlt1,type,
    c_2Ebag_2Emlt1: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > tyop_2Enum_2Enum ) > $o ) ).

thf(c_2Ecombin_2Eo,type,
    c_2Ecombin_2Eo: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).

thf(c_2Erelation_2Etransitive,type,
    c_2Erelation_2Etransitive: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $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_2Ebag_2EEMPTY__BAG,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebag_2EEMPTY__BAG @ A_27a )
      = ( c_2Ecombin_2EK @ tyop_2Enum_2Enum @ A_27a @ c_2Enum_2E0 ) ) ).

thf(thm_2Ebag_2EBAG__INN,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1n: tyop_2Enum_2Enum,V2b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__INN @ A_27a @ V0e @ V1n @ V2b )
      = ( c_2Earithmetic_2E_3E_3D @ ( V2b @ V0e ) @ V1n ) ) ).

thf(thm_2Ebag_2ESUB__BAG,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 )
    <=> ! [V2x: A_27a,V3n: tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__INN @ A_27a @ V2x @ V3n @ V0b1 )
         => ( c_2Ebag_2EBAG__INN @ A_27a @ V2x @ V3n @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EPSUB__BAG,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EPSUB__BAG @ A_27a @ V0b1 @ V1b2 )
    <=> ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 )
        & ( (~) @ ( V0b1 = V1b2 ) ) ) ) ).

thf(thm_2Ebag_2EBAG__IN,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V1b )
      = ( c_2Ebag_2EBAG__INN @ A_27a @ V0e @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1b ) ) ).

thf(thm_2Ebag_2EBAG__UNION,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1c: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b @ V1c )
      = ( ^ [V2x: A_27a] : ( c_2Earithmetic_2E_2B @ ( V0b @ V2x ) @ ( V1c @ V2x ) ) ) ) ).

thf(thm_2Ebag_2EBAG__DIFF,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0b1 @ V1b2 )
      = ( ^ [V2x: A_27a] : ( c_2Earithmetic_2E_2D @ ( V0b1 @ V2x ) @ ( V1b2 @ V2x ) ) ) ) ).

thf(thm_2Ebag_2EBAG__INSERT,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V1b )
      = ( ^ [V2x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ A_27a @ V2x @ V0e ) @ ( c_2Earithmetic_2E_2B @ ( V1b @ V0e ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( V1b @ V2x ) ) ) ) ).

thf(thm_2Ebag_2EBAG__INTER,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__INTER @ A_27a @ V0b1 @ V1b2 )
      = ( ^ [V2x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ ( V0b1 @ V2x ) @ ( V1b2 @ V2x ) ) @ ( V0b1 @ V2x ) @ ( V1b2 @ V2x ) ) ) ) ).

thf(thm_2Ebag_2EBAG__MERGE,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__MERGE @ A_27a @ V0b1 @ V1b2 )
      = ( ^ [V2x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ ( V0b1 @ V2x ) @ ( V1b2 @ V2x ) ) @ ( V1b2 @ V2x ) @ ( V0b1 @ V2x ) ) ) ) ).

thf(thm_2Ebag_2EBAG__DELETE,axiom,
    ! [A_27a: $tType,V0b0: A_27a > tyop_2Enum_2Enum,V1e: A_27a,V2b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b0 @ V1e @ V2b )
    <=> ( V0b0
        = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1e @ V2b ) ) ) ).

thf(thm_2Ebag_2ESING__BAG,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESING__BAG @ A_27a @ V0b )
    <=> ? [V1x: A_27a] :
          ( V0b
          = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1x @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).

thf(thm_2Ebag_2EEL__BAG,axiom,
    ! [A_27a: $tType,V0e: A_27a] :
      ( ( c_2Ebag_2EEL__BAG @ A_27a @ V0e )
      = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ).

thf(thm_2Ebag_2ESET__OF__BAG,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b )
      = ( ^ [V1x: A_27a] : ( c_2Ebag_2EBAG__IN @ A_27a @ V1x @ V0b ) ) ) ).

thf(thm_2Ebag_2EBAG__OF__SET,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o] :
      ( ( c_2Ebag_2EBAG__OF__SET @ A_27a @ V0P )
      = ( ^ [V1x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V0P ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ c_2Enum_2E0 ) ) ) ).

thf(thm_2Ebag_2EBAG__DISJOINT,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ V0b1 @ V1b2 )
      = ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b1 ) @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EFINITE__BAG,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
    <=> ! [V1P: ( A_27a > tyop_2Enum_2Enum ) > $o] :
          ( ( ( V1P @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
            & ! [V2b: A_27a > tyop_2Enum_2Enum] :
                ( ( V1P @ V2b )
               => ! [V3e: A_27a] : ( V1P @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V3e @ V2b ) ) ) )
         => ( V1P @ V0b ) ) ) ).

thf(thm_2Ebag_2EBAG__CARD__RELn,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__CARD__RELn @ A_27a @ V0b @ V1n )
    <=> ! [V2P: ( A_27a > tyop_2Enum_2Enum ) > tyop_2Enum_2Enum > $o] :
          ( ( ( V2P @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) @ c_2Enum_2E0 )
            & ! [V3b: A_27a > tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
                ( ( V2P @ V3b @ V4n )
               => ! [V5e: A_27a] : ( V2P @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V5e @ V3b ) @ ( c_2Enum_2ESUC @ V4n ) ) ) )
         => ( V2P @ V0b @ V1n ) ) ) ).

thf(thm_2Ebag_2EBAG__CARD,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
     => ( c_2Ebag_2EBAG__CARD__RELn @ A_27a @ V0b @ ( c_2Ebag_2EBAG__CARD @ A_27a @ V0b ) ) ) ).

thf(thm_2Ebag_2EBAG__FILTER__DEF,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__FILTER @ A_27a @ V0P @ V1b )
      = ( ^ [V2e: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( V0P @ V2e ) @ ( V1b @ V2e ) @ c_2Enum_2E0 ) ) ) ).

thf(thm_2Ebag_2EBAG__IMAGE__DEF,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27b > A_27a,V1b: A_27b > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27b @ V0f @ V1b )
      = ( ^ [V2e: A_27a] :
            ( c_2Ebool_2ELET @ ( A_27b > tyop_2Enum_2Enum ) @ tyop_2Enum_2Enum
            @ ^ [V3sb: A_27b > tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebag_2EFINITE__BAG @ A_27b @ V3sb ) @ ( c_2Ebag_2EBAG__CARD @ A_27b @ V3sb ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
            @ ( c_2Ebag_2EBAG__FILTER @ A_27b
              @ ^ [V4e0: A_27b] : ( c_2Emin_2E_3D @ A_27a @ ( V0f @ V4e0 ) @ V2e )
              @ V1b ) ) ) ) ).

thf(thm_2Ebag_2EBAG__CHOICE__DEF,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( (~)
        @ ( V0b
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
     => ( c_2Ebag_2EBAG__IN @ A_27a @ ( c_2Ebag_2EBAG__CHOICE @ A_27a @ V0b ) @ V0b ) ) ).

thf(thm_2Ebag_2EBAG__REST__DEF,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__REST @ A_27a @ V0b )
      = ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0b @ ( c_2Ebag_2EEL__BAG @ A_27a @ ( c_2Ebag_2EBAG__CHOICE @ A_27a @ V0b ) ) ) ) ).

thf(thm_2Ebag_2EBAG__GEN__SUM__def,axiom,
    ! [V0bag: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__GEN__SUM @ V0bag @ V1n )
      = ( c_2Ebag_2EITBAG @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Earithmetic_2E_2B @ V0bag @ V1n ) ) ).

thf(thm_2Ebag_2EBAG__GEN__PROD__def,axiom,
    ! [V0bag: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__GEN__PROD @ V0bag @ V1n )
      = ( c_2Ebag_2EITBAG @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Earithmetic_2E_2A @ V0bag @ V1n ) ) ).

thf(thm_2Ebag_2EBAG__EVERY,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__EVERY @ A_27a @ V0P @ V1b )
    <=> ! [V2e: A_27a] :
          ( ( c_2Ebag_2EBAG__IN @ A_27a @ V2e @ V1b )
         => ( V0P @ V2e ) ) ) ).

thf(thm_2Ebag_2EBAG__ALL__DISTINCT,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V0b )
    <=> ! [V1e: A_27a] : ( c_2Earithmetic_2E_3C_3D @ ( V0b @ V1e ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).

thf(thm_2Ebag_2EBIG__BAG__UNION__def,axiom,
    ! [A_27a: $tType,V0sob: ( A_27a > tyop_2Enum_2Enum ) > $o] :
      ( ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ V0sob )
      = ( ^ [V1x: A_27a] :
            ( c_2Epred__set_2ESUM__IMAGE @ ( A_27a > tyop_2Enum_2Enum )
            @ ^ [V2b: A_27a > tyop_2Enum_2Enum] : ( V2b @ V1x )
            @ V0sob ) ) ) ).

thf(thm_2Ebag_2Emlt1__def,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2Emlt1 @ A_27a @ V0r @ V1b1 @ V2b2 )
    <=> ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b1 )
        & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V2b2 )
        & ? [V3e: A_27a,V4rep: A_27a > tyop_2Enum_2Enum,V5res: A_27a > tyop_2Enum_2Enum] :
            ( ( V1b1
              = ( c_2Ebag_2EBAG__UNION @ A_27a @ V4rep @ V5res ) )
            & ( V2b2
              = ( c_2Ebag_2EBAG__UNION @ A_27a @ V5res @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V3e @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) )
            & ! [V6e_27: A_27a] :
                ( ( c_2Ebag_2EBAG__IN @ A_27a @ V6e_27 @ V4rep )
               => ( V0r @ V6e_27 @ V3e ) ) ) ) ) ).

thf(thm_2Ebag_2Edominates__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1s1: A_27a > $o,V2s2: A_27b > $o] :
      ( ( c_2Ebag_2Edominates @ A_27a @ A_27b @ V0R @ V1s1 @ V2s2 )
    <=> ! [V3x: A_27a] :
          ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s1 )
         => ? [V4y: A_27b] :
              ( ( c_2Ebool_2EIN @ A_27b @ V4y @ V2s2 )
              & ( V0R @ V3x @ V4y ) ) ) ) ).

thf(thm_2Ebag_2Ebag__size__def,axiom,
    ! [A_27a: $tType,V0eltsize: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2Ebag__size @ A_27a @ V0eltsize @ V1b )
      = ( c_2Ebag_2EITBAG @ A_27a @ tyop_2Enum_2Enum
        @ ^ [V2e: A_27a,V3acc: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( V0eltsize @ V2e ) ) @ V3acc )
        @ V1b
        @ c_2Enum_2E0 ) ) ).

thf(thm_2Ebag_2EEMPTY__BAG__alt,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebag_2EEMPTY__BAG @ A_27a )
      = ( ^ [V0x: A_27a] : c_2Enum_2E0 ) ) ).

thf(thm_2Ebag_2EBAG__cases,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( V0b
        = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      | ? [V1b0: A_27a > tyop_2Enum_2Enum,V2e: A_27a] :
          ( V0b
          = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2e @ V1b0 ) ) ) ).

thf(thm_2Ebag_2EBAG__MERGE__IDEM,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__MERGE @ A_27a @ V0b @ V0b )
      = V0b ) ).

thf(thm_2Ebag_2EBAG__INN__0,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1e: A_27a] : ( c_2Ebag_2EBAG__INN @ A_27a @ V1e @ c_2Enum_2E0 @ V0b ) ).

thf(thm_2Ebag_2EBAG__INN__LESS,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1e: A_27a,V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__INN @ A_27a @ V1e @ V2n @ V0b )
        & ( c_2Eprim__rec_2E_3C @ V3m @ V2n ) )
     => ( c_2Ebag_2EBAG__INN @ A_27a @ V1e @ V3m @ V0b ) ) ).

thf(thm_2Ebag_2EBAG__IN__BAG__INSERT,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1e1: A_27a,V2e2: A_27a] :
      ( ( c_2Ebag_2EBAG__IN @ A_27a @ V1e1 @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2e2 @ V0b ) )
    <=> ( ( V1e1 = V2e2 )
        | ( c_2Ebag_2EBAG__IN @ A_27a @ V1e1 @ V0b ) ) ) ).

thf(thm_2Ebag_2EBAG__INN__BAG__INSERT,axiom,
    ! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum,V2e1: A_27a,V3e2: A_27a] :
      ( ( c_2Ebag_2EBAG__INN @ A_27a @ V2e1 @ V0n @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V3e2 @ V1b ) )
    <=> ( ( ( c_2Ebag_2EBAG__INN @ A_27a @ V2e1 @ ( c_2Earithmetic_2E_2D @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1b )
          & ( V2e1 = V3e2 ) )
        | ( c_2Ebag_2EBAG__INN @ A_27a @ V2e1 @ V0n @ V1b ) ) ) ).

thf(thm_2Ebag_2EBAG__INN__BAG__INSERT__STRONG,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2e1: A_27a,V3e2: A_27a] :
      ( ( c_2Ebag_2EBAG__INN @ A_27a @ V2e1 @ V1n @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V3e2 @ V0b ) )
    <=> ( ( ( c_2Ebag_2EBAG__INN @ A_27a @ V2e1 @ ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0b )
          & ( V2e1 = V3e2 ) )
        | ( ( c_2Ebag_2EBAG__INN @ A_27a @ V2e1 @ V1n @ V0b )
          & ( (~) @ ( V2e1 = V3e2 ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__UNION__EQ__LCANCEL1,axiom,
    ! [A_27a: $tType,V0c: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( V1b
        = ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b @ V0c ) )
    <=> ( V0c
        = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ).

thf(thm_2Ebag_2EBAG__UNION__EQ__RCANCEL1,axiom,
    ! [A_27a: $tType,V0c: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( V1b
        = ( c_2Ebag_2EBAG__UNION @ A_27a @ V0c @ V1b ) )
    <=> ( V0c
        = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ).

thf(thm_2Ebag_2EBAG__IN__BAG__UNION,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2e: A_27a] :
      ( ( c_2Ebag_2EBAG__IN @ A_27a @ V2e @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) )
    <=> ( ( c_2Ebag_2EBAG__IN @ A_27a @ V2e @ V0b1 )
        | ( c_2Ebag_2EBAG__IN @ A_27a @ V2e @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__INN__BAG__UNION,axiom,
    ! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1e: A_27a,V2b1: A_27a > tyop_2Enum_2Enum,V3b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__INN @ A_27a @ V1e @ V0n @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V2b1 @ V3b2 ) )
    <=> ? [V4m1: tyop_2Enum_2Enum,V5m2: tyop_2Enum_2Enum] :
          ( ( V0n
            = ( c_2Earithmetic_2E_2B @ V4m1 @ V5m2 ) )
          & ( c_2Ebag_2EBAG__INN @ A_27a @ V1e @ V4m1 @ V2b1 )
          & ( c_2Ebag_2EBAG__INN @ A_27a @ V1e @ V5m2 @ V3b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__INN__BAG__MERGE,axiom,
    ! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1e: A_27a,V2b1: A_27a > tyop_2Enum_2Enum,V3b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__INN @ A_27a @ V1e @ V0n @ ( c_2Ebag_2EBAG__MERGE @ A_27a @ V2b1 @ V3b2 ) )
    <=> ( ( c_2Ebag_2EBAG__INN @ A_27a @ V1e @ V0n @ V2b1 )
        | ( c_2Ebag_2EBAG__INN @ A_27a @ V1e @ V0n @ V3b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__IN__BAG__MERGE,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ ( c_2Ebag_2EBAG__MERGE @ A_27a @ V1b1 @ V2b2 ) )
    <=> ( ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V1b1 )
        | ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V2b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__EXTENSION,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( V0b1 = V1b2 )
    <=> ! [V2n: tyop_2Enum_2Enum,V3e: A_27a] :
          ( ( c_2Ebag_2EBAG__INN @ A_27a @ V3e @ V2n @ V0b1 )
          = ( c_2Ebag_2EBAG__INN @ A_27a @ V3e @ V2n @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__UNION__INSERT,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V1b1 ) @ V2b2 )
        = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b1 @ V2b2 ) ) )
      & ( ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b1 @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V2b2 ) )
        = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b1 @ V2b2 ) ) ) ) ).

thf(thm_2Ebag_2EBAG__INSERT__DIFF,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1b: A_27a > tyop_2Enum_2Enum] :
      ( (~)
      @ ( ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ V1b )
        = V1b ) ) ).

thf(thm_2Ebag_2EBAG__INSERT__NOT__EMPTY,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1b: A_27a > tyop_2Enum_2Enum] :
      ( (~)
      @ ( ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ V1b )
        = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ).

thf(thm_2Ebag_2EBAG__INSERT__ONE__ONE,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2x: A_27a] :
      ( ( ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2x @ V0b1 )
        = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2x @ V1b2 ) )
    <=> ( V0b1 = V1b2 ) ) ).

thf(thm_2Ebag_2EC__BAG__INSERT__ONE__ONE,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1y: A_27a,V2b: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ V2b )
        = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1y @ V2b ) )
    <=> ( V0x = V1y ) ) ).

thf(thm_2Ebag_2EBAG__INSERT__commutes,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1e1: A_27a,V2e2: A_27a] :
      ( ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1e1 @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2e2 @ V0b ) )
      = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2e2 @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1e1 @ V0b ) ) ) ).

thf(thm_2Ebag_2EBAG__DECOMPOSE,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V1b )
     => ? [V2b_27: A_27a > tyop_2Enum_2Enum] :
          ( V1b
          = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V2b_27 ) ) ) ).

thf(thm_2Ebag_2EBAG__UNION__LEFT__CANCEL,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b @ V1b1 )
        = ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b @ V2b2 ) )
    <=> ( V1b1 = V2b2 ) ) ).

thf(thm_2Ebag_2ECOMM__BAG__UNION,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 )
      = ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b2 @ V0b1 ) ) ).

thf(thm_2Ebag_2EBAG__UNION__RIGHT__CANCEL,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b1 @ V0b )
        = ( c_2Ebag_2EBAG__UNION @ A_27a @ V2b2 @ V0b ) )
    <=> ( V1b1 = V2b2 ) ) ).

thf(thm_2Ebag_2EASSOC__BAG__UNION,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2b3: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b2 @ V2b3 ) )
      = ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) @ V2b3 ) ) ).

thf(thm_2Ebag_2EBAG__UNION__EMPTY,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType] :
      ( ! [V0b: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
          = V0b )
      & ! [V1b: A_27b > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__UNION @ A_27b @ ( c_2Ebag_2EEMPTY__BAG @ A_27b ) @ V1b )
          = V1b )
      & ! [V2b1: A_27c > tyop_2Enum_2Enum,V3b2: A_27c > tyop_2Enum_2Enum] :
          ( ( ( c_2Ebag_2EBAG__UNION @ A_27c @ V2b1 @ V3b2 )
            = ( c_2Ebag_2EEMPTY__BAG @ A_27c ) )
        <=> ( ( V2b1
              = ( c_2Ebag_2EEMPTY__BAG @ A_27c ) )
            & ( V3b2
              = ( c_2Ebag_2EEMPTY__BAG @ A_27c ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__DELETE__EMPTY,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b: A_27a > tyop_2Enum_2Enum] : ( (~) @ ( c_2Ebag_2EBAG__DELETE @ A_27a @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) @ V0e @ V1b ) ) ).

thf(thm_2Ebag_2EBAG__DELETE__commutes,axiom,
    ! [A_27a: $tType,V0b0: A_27a > tyop_2Enum_2Enum,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum,V3e1: A_27a,V4e2: A_27a] :
      ( ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b0 @ V3e1 @ V1b1 )
        & ( c_2Ebag_2EBAG__DELETE @ A_27a @ V1b1 @ V4e2 @ V2b2 ) )
     => ? [V5b_27: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b0 @ V4e2 @ V5b_27 )
          & ( c_2Ebag_2EBAG__DELETE @ A_27a @ V5b_27 @ V3e1 @ V2b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__DELETE__11,axiom,
    ! [A_27a: $tType,V0b0: A_27a > tyop_2Enum_2Enum,V1e1: A_27a,V2e2: A_27a,V3b1: A_27a > tyop_2Enum_2Enum,V4b2: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b0 @ V1e1 @ V3b1 )
        & ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b0 @ V2e2 @ V4b2 ) )
     => ( ( V1e1 = V2e2 )
      <=> ( V3b1 = V4b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__INN__BAG__DELETE,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2e: A_27a] :
      ( ( ( c_2Ebag_2EBAG__INN @ A_27a @ V2e @ V1n @ V0b )
        & ( c_2Earithmetic_2E_3E @ V1n @ c_2Enum_2E0 ) )
     => ? [V3b_27: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b @ V2e @ V3b_27 ) ) ).

thf(thm_2Ebag_2EBAG__IN__BAG__DELETE,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1e: A_27a] :
      ( ( c_2Ebag_2EBAG__IN @ A_27a @ V1e @ V0b )
     => ? [V2b_27: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b @ V1e @ V2b_27 ) ) ).

thf(thm_2Ebag_2EBAG__DELETE__INSERT,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1y: A_27a,V2b1: A_27a > tyop_2Enum_2Enum,V3b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ V2b1 ) @ V1y @ V3b2 )
     => ( ( ( V0x = V1y )
          & ( V2b1 = V3b2 ) )
        | ( ? [V4b3: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2EBAG__DELETE @ A_27a @ V2b1 @ V1y @ V4b3 )
          & ( (~) @ ( V0x = V1y ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__DELETE__BAG__IN__up,axiom,
    ! [A_27a: $tType,V0b0: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum,V2e: A_27a] :
      ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b0 @ V2e @ V1b )
     => ! [V3e_27: A_27a] :
          ( ( c_2Ebag_2EBAG__IN @ A_27a @ V3e_27 @ V1b )
         => ( c_2Ebag_2EBAG__IN @ A_27a @ V3e_27 @ V0b0 ) ) ) ).

thf(thm_2Ebag_2EBAG__DELETE__BAG__IN__down,axiom,
    ! [A_27a: $tType,V0b0: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum,V2e1: A_27a,V3e2: A_27a] :
      ( ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b0 @ V2e1 @ V1b )
        & ( (~) @ ( V2e1 = V3e2 ) )
        & ( c_2Ebag_2EBAG__IN @ A_27a @ V3e2 @ V0b0 ) )
     => ( c_2Ebag_2EBAG__IN @ A_27a @ V3e2 @ V1b ) ) ).

thf(thm_2Ebag_2EBAG__DELETE__BAG__IN,axiom,
    ! [A_27a: $tType,V0b0: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum,V2e: A_27a] :
      ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b0 @ V2e @ V1b )
     => ( c_2Ebag_2EBAG__IN @ A_27a @ V2e @ V0b0 ) ) ).

thf(thm_2Ebag_2EBAG__DELETE__concrete,axiom,
    ! [A_27a: $tType,V0b0: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum,V2e: A_27a] :
      ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b0 @ V2e @ V1b )
    <=> ( ( c_2Earithmetic_2E_3E @ ( V0b0 @ V2e ) @ c_2Enum_2E0 )
        & ( V1b
          = ( ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V2e ) @ ( c_2Earithmetic_2E_2D @ ( V0b0 @ V2e ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( V0b0 @ V3x ) ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__UNION__DIFF__eliminate,axiom,
    ! [A_27a: $tType,V0c: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b @ V0c ) @ V0c )
        = V1b )
      & ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0c @ V1b ) @ V0c )
        = V1b ) ) ).

thf(thm_2Ebag_2EBAG__INSERT__EQUAL,axiom,
    ! [A_27a: $tType,V0b: A_27a,V1a: A_27a,V2N: A_27a > tyop_2Enum_2Enum,V3M: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1a @ V3M )
        = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0b @ V2N ) )
    <=> ( ( ( V3M = V2N )
          & ( V1a = V0b ) )
        | ? [V4k: A_27a > tyop_2Enum_2Enum] :
            ( ( V3M
              = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0b @ V4k ) )
            & ( V2N
              = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1a @ V4k ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__DELETE__TWICE,axiom,
    ! [A_27a: $tType,V0b0: A_27a > tyop_2Enum_2Enum,V1e1: A_27a,V2e2: A_27a,V3b1: A_27a > tyop_2Enum_2Enum,V4b2: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b0 @ V1e1 @ V3b1 )
        & ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b0 @ V2e2 @ V4b2 )
        & ( (~) @ ( V3b1 = V4b2 ) ) )
     => ? [V5b: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ V3b1 @ V2e2 @ V5b )
          & ( c_2Ebag_2EBAG__DELETE @ A_27a @ V4b2 @ V1e1 @ V5b ) ) ) ).

thf(thm_2Ebag_2ESING__BAG__THM,axiom,
    ! [A_27a: $tType,V0e: A_27a] : ( c_2Ebag_2ESING__BAG @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ).

thf(thm_2Ebag_2EEL__BAG__11,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
      ( ( ( c_2Ebag_2EEL__BAG @ A_27a @ V0x )
        = ( c_2Ebag_2EEL__BAG @ A_27a @ V1y ) )
     => ( V0x = V1y ) ) ).

thf(thm_2Ebag_2EEL__BAG__INSERT__squeeze,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1b: A_27a > tyop_2Enum_2Enum,V2y: A_27a] :
      ( ( ( c_2Ebag_2EEL__BAG @ A_27a @ V0x )
        = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2y @ V1b ) )
     => ( ( V0x = V2y )
        & ( V1b
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).

thf(thm_2Ebag_2ESING__EL__BAG,axiom,
    ! [A_27a: $tType,V0e: A_27a] : ( c_2Ebag_2ESING__BAG @ A_27a @ ( c_2Ebag_2EEL__BAG @ A_27a @ V0e ) ) ).

thf(thm_2Ebag_2EBAG__INSERT__UNION,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1e: A_27a] :
      ( ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1e @ V0b )
      = ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EEL__BAG @ A_27a @ V1e ) @ V0b ) ) ).

thf(thm_2Ebag_2EBAG__INSERT__EQ__UNION,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum,V3b: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V3b )
        = ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b1 @ V2b2 ) )
     => ( ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V1b1 )
        | ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V2b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__DELETE__SING,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1e: A_27a] :
      ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b @ V1e @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
     => ( c_2Ebag_2ESING__BAG @ A_27a @ V0b ) ) ).

thf(thm_2Ebag_2ENOT__IN__EMPTY__BAG,axiom,
    ! [A_27a: $tType,V0x: A_27a] : ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27a @ V0x @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ).

thf(thm_2Ebag_2EBAG__INN__EMPTY__BAG,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__INN @ A_27a @ V0e @ V1n @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
    <=> ( V1n = c_2Enum_2E0 ) ) ).

thf(thm_2Ebag_2EMEMBER__NOT__EMPTY,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ? [V1x: A_27a] : ( c_2Ebag_2EBAG__IN @ A_27a @ V1x @ V0b )
    <=> ( (~)
        @ ( V0b
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__LEQ,axiom,
    ! [A_27a: $tType,V0b2: A_27a > tyop_2Enum_2Enum,V1b1: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b1 @ V0b2 )
    <=> ! [V2x: A_27a] : ( c_2Earithmetic_2E_3C_3D @ ( V1b1 @ V2x ) @ ( V0b2 @ V2x ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__EMPTY,axiom,
    ! [A_27a: $tType] :
      ( ! [V0b: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) @ V0b )
      & ! [V1b: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
        <=> ( V1b
            = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__REFL,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b @ V0b ) ).

thf(thm_2Ebag_2EPSUB__BAG__IRREFL,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] : ( (~) @ ( c_2Ebag_2EPSUB__BAG @ A_27a @ V0b @ V0b ) ) ).

thf(thm_2Ebag_2ESUB__BAG__ANTISYM,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 )
        & ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b2 @ V0b1 ) )
     => ( V0b1 = V1b2 ) ) ).

thf(thm_2Ebag_2EPSUB__BAG__ANTISYM,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( (~)
      @ ( ( c_2Ebag_2EPSUB__BAG @ A_27a @ V0b1 @ V1b2 )
        & ( c_2Ebag_2EPSUB__BAG @ A_27a @ V1b2 @ V0b1 ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__TRANS,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2b3: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 )
        & ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b2 @ V2b3 ) )
     => ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V2b3 ) ) ).

thf(thm_2Ebag_2EPSUB__BAG__TRANS,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2b3: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EPSUB__BAG @ A_27a @ V0b1 @ V1b2 )
        & ( c_2Ebag_2EPSUB__BAG @ A_27a @ V1b2 @ V2b3 ) )
     => ( c_2Ebag_2EPSUB__BAG @ A_27a @ V0b1 @ V2b3 ) ) ).

thf(thm_2Ebag_2EPSUB__BAG__SUB__BAG,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EPSUB__BAG @ A_27a @ V0b1 @ V1b2 )
     => ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 ) ) ).

thf(thm_2Ebag_2EPSUB__BAG__NOT__EQ,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EPSUB__BAG @ A_27a @ V0b1 @ V1b2 )
     => ( (~) @ ( V0b1 = V1b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__DIFF__EMPTY,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] :
      ( ! [V0b: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0b @ V0b )
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      & ! [V1b: A_27b > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__DIFF @ A_27b @ V1b @ ( c_2Ebag_2EEMPTY__BAG @ A_27b ) )
          = V1b )
      & ! [V2b: A_27c > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__DIFF @ A_27c @ ( c_2Ebag_2EEMPTY__BAG @ A_27c ) @ V2b )
          = ( c_2Ebag_2EEMPTY__BAG @ A_27c ) )
      & ! [V3b1: A_27d > tyop_2Enum_2Enum,V4b2: A_27d > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27d @ V3b1 @ V4b2 )
         => ( ( c_2Ebag_2EBAG__DIFF @ A_27d @ V3b1 @ V4b2 )
            = ( c_2Ebag_2EEMPTY__BAG @ A_27d ) ) ) ) ).

thf(thm_2Ebag_2EBAG__DIFF__EMPTY__simple,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType] :
      ( ! [V0b: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0b @ V0b )
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      & ! [V1b: A_27b > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__DIFF @ A_27b @ V1b @ ( c_2Ebag_2EEMPTY__BAG @ A_27b ) )
          = V1b )
      & ! [V2b: A_27c > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__DIFF @ A_27c @ ( c_2Ebag_2EEMPTY__BAG @ A_27c ) @ V2b )
          = ( c_2Ebag_2EEMPTY__BAG @ A_27c ) ) ) ).

thf(thm_2Ebag_2EBAG__DIFF__EQ__EMPTY,axiom,
    ! [A_27a: $tType,V0c: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1b @ V0c )
        = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
    <=> ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b @ V0c ) ) ).

thf(thm_2Ebag_2EBAG__DIFF__INSERT__same,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ V1b1 ) @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ V2b2 ) )
      = ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1b1 @ V2b2 ) ) ).

thf(thm_2Ebag_2EBAG__DIFF__INSERT,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
      ( ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27a @ V0x @ V1b1 ) )
     => ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ V2b2 ) @ V1b1 )
        = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V2b2 @ V1b1 ) ) ) ) ).

thf(thm_2Ebag_2ENOT__IN__BAG__DIFF,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
      ( ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27a @ V0x @ V1b1 ) )
     => ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1b1 @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ V2b2 ) )
        = ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1b1 @ V2b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__IN__DIFF__I,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b2: A_27a > tyop_2Enum_2Enum,V2b1: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V2b1 )
        & ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V1b2 ) ) )
     => ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V2b1 @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__IN__DIFF__E,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b2: A_27a > tyop_2Enum_2Enum,V2b1: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V2b1 @ V1b2 ) )
     => ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V2b1 ) ) ).

thf(thm_2Ebag_2EBAG__UNION__DIFF,axiom,
    ! [A_27a: $tType,V0X: A_27a > tyop_2Enum_2Enum,V1Y: A_27a > tyop_2Enum_2Enum,V2Z: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V2Z @ V1Y )
     => ( ( ( c_2Ebag_2EBAG__UNION @ A_27a @ V0X @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1Y @ V2Z ) )
          = ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0X @ V1Y ) @ V2Z ) )
        & ( ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1Y @ V2Z ) @ V0X )
          = ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0X @ V1Y ) @ V2Z ) ) ) ) ).

thf(thm_2Ebag_2EBAG__DIFF__2L,axiom,
    ! [A_27a: $tType,V0X: A_27a > tyop_2Enum_2Enum,V1Y: A_27a > tyop_2Enum_2Enum,V2Z: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0X @ V1Y ) @ V2Z )
      = ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0X @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1Y @ V2Z ) ) ) ).

thf(thm_2Ebag_2EBAG__DIFF__2R,axiom,
    ! [A_27a: $tType,V0A: A_27a > tyop_2Enum_2Enum,V1B: A_27a > tyop_2Enum_2Enum,V2C: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V2C @ V1B )
     => ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0A @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1B @ V2C ) )
        = ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0A @ V2C ) @ V1B ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__BAG__DIFF,axiom,
    ! [A_27a: $tType,V0X: A_27a > tyop_2Enum_2Enum,V1Y: A_27a > tyop_2Enum_2Enum,V2Y_27: A_27a > tyop_2Enum_2Enum,V3Z: A_27a > tyop_2Enum_2Enum,V4W: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0X @ V1Y ) @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V3Z @ V4W ) )
     => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0X @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1Y @ V2Y_27 ) ) @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V3Z @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V4W @ V2Y_27 ) ) ) ) ).

thf(thm_2Ebag_2EBAG__DIFF__UNION__eliminate,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2b3: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V2b3 ) )
        = ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1b2 @ V2b3 ) )
      & ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V2b3 @ V0b1 ) )
        = ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1b2 @ V2b3 ) )
      & ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b2 @ V0b1 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V2b3 ) )
        = ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1b2 @ V2b3 ) )
      & ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b2 @ V0b1 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V2b3 @ V0b1 ) )
        = ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1b2 @ V2b3 ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__UNION__eliminate,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2b3: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V2b3 ) )
        = ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b2 @ V2b3 ) )
      & ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V2b3 @ V0b1 ) )
        = ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b2 @ V2b3 ) )
      & ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b2 @ V0b1 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V2b3 ) )
        = ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b2 @ V2b3 ) )
      & ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b2 @ V0b1 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V2b3 @ V0b1 ) )
        = ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b2 @ V2b3 ) ) ) ).

thf(thm_2Ebag_2Emove__BAG__UNION__over__eq,axiom,
    ! [A_27a: $tType,V0X: A_27a > tyop_2Enum_2Enum,V1Y: A_27a > tyop_2Enum_2Enum,V2Z: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__UNION @ A_27a @ V0X @ V1Y )
        = V2Z )
     => ( V0X
        = ( c_2Ebag_2EBAG__DIFF @ A_27a @ V2Z @ V1Y ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__UNION,axiom,
    ! [A_27a: $tType] :
      ( ! [V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 )
         => ! [V2b: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b2 @ V2b ) ) )
      & ! [V3b1: A_27a > tyop_2Enum_2Enum,V4b2: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V3b1 @ V4b2 )
         => ! [V5b: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ V3b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V5b @ V4b2 ) ) )
      & ! [V6b1: A_27a > tyop_2Enum_2Enum,V7b2: A_27a > tyop_2Enum_2Enum,V8b3: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V6b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V7b2 @ V8b3 ) )
         => ! [V9b: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ V6b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V7b2 @ V9b ) @ V8b3 ) ) )
      & ! [V10b1: A_27a > tyop_2Enum_2Enum,V11b2: A_27a > tyop_2Enum_2Enum,V12b3: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V10b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V11b2 @ V12b3 ) )
         => ! [V13b: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ V10b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V13b @ V11b2 ) @ V12b3 ) ) )
      & ! [V14b1: A_27a > tyop_2Enum_2Enum,V15b2: A_27a > tyop_2Enum_2Enum,V16b3: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V14b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V16b3 @ V15b2 ) )
         => ! [V17b: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ V14b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V16b3 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V15b2 @ V17b ) ) ) )
      & ! [V18b1: A_27a > tyop_2Enum_2Enum,V19b2: A_27a > tyop_2Enum_2Enum,V20b3: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V18b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V20b3 @ V19b2 ) )
         => ! [V21b: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ V18b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V20b3 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V21b @ V19b2 ) ) ) )
      & ! [V22b1: A_27a > tyop_2Enum_2Enum,V23b2: A_27a > tyop_2Enum_2Enum,V24b3: A_27a > tyop_2Enum_2Enum,V25b4: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V22b1 @ V24b3 )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V23b2 @ V25b4 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V22b1 @ V23b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V24b3 @ V25b4 ) ) ) )
      & ! [V26b1: A_27a > tyop_2Enum_2Enum,V27b2: A_27a > tyop_2Enum_2Enum,V28b3: A_27a > tyop_2Enum_2Enum,V29b4: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V26b1 @ V29b4 )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V27b2 @ V28b3 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V26b1 @ V27b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V28b3 @ V29b4 ) ) ) )
      & ! [V30b1: A_27a > tyop_2Enum_2Enum,V31b2: A_27a > tyop_2Enum_2Enum,V32b3: A_27a > tyop_2Enum_2Enum,V33b4: A_27a > tyop_2Enum_2Enum,V34b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V30b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V32b3 @ V34b5 ) )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V31b2 @ V33b4 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V30b1 @ V31b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V32b3 @ V33b4 ) @ V34b5 ) ) ) )
      & ! [V35b1: A_27a > tyop_2Enum_2Enum,V36b2: A_27a > tyop_2Enum_2Enum,V37b3: A_27a > tyop_2Enum_2Enum,V38b4: A_27a > tyop_2Enum_2Enum,V39b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V35b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V38b4 @ V39b5 ) )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V36b2 @ V37b3 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V35b1 @ V36b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V37b3 @ V38b4 ) @ V39b5 ) ) ) )
      & ! [V40b1: A_27a > tyop_2Enum_2Enum,V41b2: A_27a > tyop_2Enum_2Enum,V42b3: A_27a > tyop_2Enum_2Enum,V43b4: A_27a > tyop_2Enum_2Enum,V44b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V41b2 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V42b3 @ V44b5 ) )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V40b1 @ V43b4 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V40b1 @ V41b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V42b3 @ V43b4 ) @ V44b5 ) ) ) )
      & ! [V45b1: A_27a > tyop_2Enum_2Enum,V46b2: A_27a > tyop_2Enum_2Enum,V47b3: A_27a > tyop_2Enum_2Enum,V48b4: A_27a > tyop_2Enum_2Enum,V49b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V46b2 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V48b4 @ V49b5 ) )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V45b1 @ V47b3 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V45b1 @ V46b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V47b3 @ V48b4 ) @ V49b5 ) ) ) )
      & ! [V50b1: A_27a > tyop_2Enum_2Enum,V51b2: A_27a > tyop_2Enum_2Enum,V52b3: A_27a > tyop_2Enum_2Enum,V53b4: A_27a > tyop_2Enum_2Enum,V54b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V50b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V54b5 @ V52b3 ) )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V51b2 @ V53b4 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V51b2 @ V50b1 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V54b5 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V52b3 @ V53b4 ) ) ) ) )
      & ! [V55b1: A_27a > tyop_2Enum_2Enum,V56b2: A_27a > tyop_2Enum_2Enum,V57b3: A_27a > tyop_2Enum_2Enum,V58b4: A_27a > tyop_2Enum_2Enum,V59b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V55b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V59b5 @ V58b4 ) )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V56b2 @ V57b3 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V56b2 @ V55b1 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V59b5 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V57b3 @ V58b4 ) ) ) ) )
      & ! [V60b1: A_27a > tyop_2Enum_2Enum,V61b2: A_27a > tyop_2Enum_2Enum,V62b3: A_27a > tyop_2Enum_2Enum,V63b4: A_27a > tyop_2Enum_2Enum,V64b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V61b2 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V64b5 @ V62b3 ) )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V60b1 @ V63b4 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V61b2 @ V60b1 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V64b5 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V62b3 @ V63b4 ) ) ) ) )
      & ! [V65b1: A_27a > tyop_2Enum_2Enum,V66b2: A_27a > tyop_2Enum_2Enum,V67b3: A_27a > tyop_2Enum_2Enum,V68b4: A_27a > tyop_2Enum_2Enum,V69b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V66b2 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V69b5 @ V68b4 ) )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V65b1 @ V67b3 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V66b2 @ V65b1 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V69b5 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V67b3 @ V68b4 ) ) ) ) )
      & ! [V70b1: A_27a > tyop_2Enum_2Enum,V71b2: A_27a > tyop_2Enum_2Enum,V72b3: A_27a > tyop_2Enum_2Enum,V73b4: A_27a > tyop_2Enum_2Enum,V74b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V70b1 @ V71b2 ) @ V73b4 )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V72b3 @ V74b5 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V70b1 @ V72b3 ) @ V71b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V73b4 @ V74b5 ) ) ) )
      & ! [V75b1: A_27a > tyop_2Enum_2Enum,V76b2: A_27a > tyop_2Enum_2Enum,V77b3: A_27a > tyop_2Enum_2Enum,V78b4: A_27a > tyop_2Enum_2Enum,V79b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V75b1 @ V76b2 ) @ V79b5 )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V77b3 @ V78b4 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V75b1 @ V77b3 ) @ V76b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V78b4 @ V79b5 ) ) ) )
      & ! [V80b1: A_27a > tyop_2Enum_2Enum,V81b2: A_27a > tyop_2Enum_2Enum,V82b3: A_27a > tyop_2Enum_2Enum,V83b4: A_27a > tyop_2Enum_2Enum,V84b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V82b3 @ V81b2 ) @ V83b4 )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V80b1 @ V84b5 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V80b1 @ V82b3 ) @ V81b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V83b4 @ V84b5 ) ) ) )
      & ! [V85b1: A_27a > tyop_2Enum_2Enum,V86b2: A_27a > tyop_2Enum_2Enum,V87b3: A_27a > tyop_2Enum_2Enum,V88b4: A_27a > tyop_2Enum_2Enum,V89b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V87b3 @ V86b2 ) @ V89b5 )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V85b1 @ V88b4 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V85b1 @ V87b3 ) @ V86b2 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V88b4 @ V89b5 ) ) ) )
      & ! [V90b1: A_27a > tyop_2Enum_2Enum,V91b2: A_27a > tyop_2Enum_2Enum,V92b3: A_27a > tyop_2Enum_2Enum,V93b4: A_27a > tyop_2Enum_2Enum,V94b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V91b2 @ V90b1 ) @ V93b4 )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V92b3 @ V94b5 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V91b2 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V90b1 @ V92b3 ) ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V94b5 @ V93b4 ) ) ) )
      & ! [V95b1: A_27a > tyop_2Enum_2Enum,V96b2: A_27a > tyop_2Enum_2Enum,V97b3: A_27a > tyop_2Enum_2Enum,V98b4: A_27a > tyop_2Enum_2Enum,V99b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V96b2 @ V95b1 ) @ V99b5 )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V97b3 @ V98b4 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V96b2 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V95b1 @ V97b3 ) ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V99b5 @ V98b4 ) ) ) )
      & ! [V100b1: A_27a > tyop_2Enum_2Enum,V101b2: A_27a > tyop_2Enum_2Enum,V102b3: A_27a > tyop_2Enum_2Enum,V103b4: A_27a > tyop_2Enum_2Enum,V104b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V101b2 @ V102b3 ) @ V103b4 )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V100b1 @ V104b5 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V101b2 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V100b1 @ V102b3 ) ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V104b5 @ V103b4 ) ) ) )
      & ! [V105b1: A_27a > tyop_2Enum_2Enum,V106b2: A_27a > tyop_2Enum_2Enum,V107b3: A_27a > tyop_2Enum_2Enum,V108b4: A_27a > tyop_2Enum_2Enum,V109b5: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V106b2 @ V107b3 ) @ V109b5 )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V105b1 @ V108b4 )
           => ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V106b2 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V105b1 @ V107b3 ) ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V109b5 @ V108b4 ) ) ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__EL__BAG,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EEL__BAG @ A_27a @ V0e ) @ V1b )
      = ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V1b ) ) ).

thf(thm_2Ebag_2ESUB__BAG__INSERT,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V1b1 ) @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V2b2 ) )
      = ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b1 @ V2b2 ) ) ).

thf(thm_2Ebag_2ESUB__BAG__INSERT__I,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1c: A_27a > tyop_2Enum_2Enum,V2e: A_27a] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b @ V1c )
     => ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2e @ V1c ) ) ) ).

thf(thm_2Ebag_2ENOT__IN__SUB__BAG__INSERT,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2e: A_27a] :
      ( ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27a @ V2e @ V0b1 ) )
     => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2e @ V1b2 ) )
        = ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__BAG__IN,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ V1b1 ) @ V2b2 )
     => ( c_2Ebag_2EBAG__IN @ A_27a @ V0x @ V2b2 ) ) ).

thf(thm_2Ebag_2ESUB__BAG__EXISTS,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 )
    <=> ? [V2b: A_27a > tyop_2Enum_2Enum] :
          ( V1b2
          = ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V2b ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__UNION__DIFF,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2b3: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V2b3 )
     => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b2 @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V2b3 @ V0b1 ) )
        = ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) @ V2b3 ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__UNION__down,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2b3: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) @ V2b3 )
     => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V2b3 )
        & ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b2 @ V2b3 ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__DIFF,axiom,
    ! [A_27a: $tType] :
      ( ! [V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 )
         => ! [V2b3: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0b1 @ V2b3 ) @ V1b2 ) )
      & ! [V3b1: A_27a > tyop_2Enum_2Enum,V4b2: A_27a > tyop_2Enum_2Enum,V5b3: A_27a > tyop_2Enum_2Enum,V6b4: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V4b2 @ V3b1 )
         => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V6b4 @ V5b3 )
           => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V3b1 @ V4b2 ) @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V5b3 @ V6b4 ) )
              = ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V3b1 @ V6b4 ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V4b2 @ V5b3 ) ) ) ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__PSUB__BAG,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 )
    <=> ( ( c_2Ebag_2EPSUB__BAG @ A_27a @ V0b1 @ V1b2 )
        | ( V0b1 = V1b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__DELETE__PSUB__BAG,axiom,
    ! [A_27a: $tType,V0b0: A_27a > tyop_2Enum_2Enum,V1e: A_27a,V2b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__DELETE @ A_27a @ V0b0 @ V1e @ V2b )
     => ( c_2Ebag_2EPSUB__BAG @ A_27a @ V2b @ V0b0 ) ) ).

thf(thm_2Ebag_2ESET__OF__EMPTY,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebag_2EBAG__OF__SET @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
      = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ).

thf(thm_2Ebag_2EBAG__IN__BAG__OF__SET,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1p: A_27a] :
      ( ( c_2Ebag_2EBAG__IN @ A_27a @ V1p @ ( c_2Ebag_2EBAG__OF__SET @ A_27a @ V0P ) )
      = ( c_2Ebool_2EIN @ A_27a @ V1p @ V0P ) ) ).

thf(thm_2Ebag_2EBAG__OF__EMPTY,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebag_2ESET__OF__BAG @ A_27a @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      = ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).

thf(thm_2Ebag_2ESET__BAG__I,axiom,
    ! [A_27a: $tType,V0s: A_27a > $o] :
      ( ( c_2Ebag_2ESET__OF__BAG @ A_27a @ ( c_2Ebag_2EBAG__OF__SET @ A_27a @ V0s ) )
      = V0s ) ).

thf(thm_2Ebag_2ESUB__BAG__SET,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 )
     => ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b1 ) @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V1b2 ) ) ) ).

thf(thm_2Ebag_2ESET__OF__BAG__UNION,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESET__OF__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) )
      = ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b1 ) @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V1b2 ) ) ) ).

thf(thm_2Ebag_2ESET__OF__BAG__MERGE,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESET__OF__BAG @ A_27a @ ( c_2Ebag_2EBAG__MERGE @ A_27a @ V0b1 @ V1b2 ) )
      = ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b1 ) @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V1b2 ) ) ) ).

thf(thm_2Ebag_2ESET__OF__BAG__INSERT,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESET__OF__BAG @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V1b ) )
      = ( c_2Epred__set_2EINSERT @ A_27a @ V0e @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V1b ) ) ) ).

thf(thm_2Ebag_2ESET__OF__EL__BAG,axiom,
    ! [A_27a: $tType,V0e: A_27a] :
      ( ( c_2Ebag_2ESET__OF__BAG @ A_27a @ ( c_2Ebag_2EEL__BAG @ A_27a @ V0e ) )
      = ( c_2Epred__set_2EINSERT @ A_27a @ V0e @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).

thf(thm_2Ebag_2ESET__OF__BAG__DIFF__SUBSET__down,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b1 ) @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V1b2 ) ) @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0b1 @ V1b2 ) ) ) ).

thf(thm_2Ebag_2ESET__OF__BAG__DIFF__SUBSET__up,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0b1 @ V1b2 ) ) @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b1 ) ) ).

thf(thm_2Ebag_2EIN__SET__OF__BAG,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V1b ) )
      = ( c_2Ebag_2EBAG__IN @ A_27a @ V0x @ V1b ) ) ).

thf(thm_2Ebag_2ESET__OF__BAG__EQ__EMPTY,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( ( ( c_2Epred__set_2EEMPTY @ A_27a )
          = ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b ) )
      <=> ( V0b
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
      & ( ( ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b )
          = ( c_2Epred__set_2EEMPTY @ A_27a ) )
      <=> ( V0b
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).

thf(thm_2Ebag_2EBAG__DISJOINT__EMPTY,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ V0b @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      & ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) @ V0b ) ) ).

thf(thm_2Ebag_2EBAG__DISJOINT__DIFF,axiom,
    ! [A_27a: $tType,V0B1: A_27a > tyop_2Enum_2Enum,V1B2: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0B1 @ V1B2 ) @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1B2 @ V0B1 ) ) ).

thf(thm_2Ebag_2EBAG__DISJOINT__BAG__IN,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ V0b1 @ V1b2 )
    <=> ! [V2e: A_27a] :
          ( ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27a @ V2e @ V0b1 ) )
          | ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27a @ V2e @ V1b2 ) ) ) ) ).

thf(thm_2Ebag_2EBAG__DISJOINT__BAG__INSERT,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ! [V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2e1: A_27a] :
          ( ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2e1 @ V0b1 ) @ V1b2 )
        <=> ( ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27a @ V2e1 @ V1b2 ) )
            & ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ V0b1 @ V1b2 ) ) )
      & ! [V3b1: A_27b > tyop_2Enum_2Enum,V4b2: A_27b > tyop_2Enum_2Enum,V5e2: A_27b] :
          ( ( c_2Ebag_2EBAG__DISJOINT @ A_27b @ V3b1 @ ( c_2Ebag_2EBAG__INSERT @ A_27b @ V5e2 @ V4b2 ) )
        <=> ( ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27b @ V5e2 @ V3b1 ) )
            & ( c_2Ebag_2EBAG__DISJOINT @ A_27b @ V3b1 @ V4b2 ) ) ) ) ).

thf(thm_2Ebag_2EBAG__DISJOINT__BAG__UNION,axiom,
    ! [A_27a: $tType,V0b3: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2b1: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ V2b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b2 @ V0b3 ) )
      <=> ( ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ V2b1 @ V1b2 )
          & ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ V2b1 @ V0b3 ) ) )
      & ( ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V2b1 @ V1b2 ) @ V0b3 )
      <=> ( ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ V2b1 @ V0b3 )
          & ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ V1b2 @ V0b3 ) ) ) ) ).

thf(thm_2Ebag_2EFINITE__EMPTY__BAG,axiom,
    ! [A_27a: $tType] : ( c_2Ebag_2EFINITE__BAG @ A_27a @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ).

thf(thm_2Ebag_2EFINITE__BAG__INSERT,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
     => ! [V1e: A_27a] : ( c_2Ebag_2EFINITE__BAG @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1e @ V0b ) ) ) ).

thf(thm_2Ebag_2EFINITE__BAG__INDUCT,axiom,
    ! [A_27a: $tType,V0P: ( A_27a > tyop_2Enum_2Enum ) > $o] :
      ( ( ( V0P @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
        & ! [V1b: A_27a > tyop_2Enum_2Enum] :
            ( ( V0P @ V1b )
           => ! [V2e: A_27a] : ( V0P @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2e @ V1b ) ) ) )
     => ! [V3b: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V3b )
         => ( V0P @ V3b ) ) ) ).

thf(thm_2Ebag_2ESTRONG__FINITE__BAG__INDUCT,axiom,
    ! [A_27a: $tType,V0P: ( A_27a > tyop_2Enum_2Enum ) > $o] :
      ( ( ( V0P @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
        & ! [V1b: A_27a > tyop_2Enum_2Enum] :
            ( ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b )
              & ( V0P @ V1b ) )
           => ! [V2e: A_27a] : ( V0P @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2e @ V1b ) ) ) )
     => ! [V3b: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V3b )
         => ( V0P @ V3b ) ) ) ).

thf(thm_2Ebag_2EFINITE__BAG__THM,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      & ! [V0e: A_27a,V1b: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V1b ) )
          = ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b ) ) ) ).

thf(thm_2Ebag_2EFINITE__BAG__DIFF,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b1 )
     => ! [V1b2: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2EFINITE__BAG @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0b1 @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EFINITE__BAG__DIFF__dual,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b1 )
     => ! [V1b2: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1b2 @ V0b1 ) )
         => ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EFINITE__BAG__UNION,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) )
    <=> ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b1 )
        & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EFINITE__SUB__BAG,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b1 )
     => ! [V1b2: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b2 @ V0b1 )
         => ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__CARD__EMPTY,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebag_2EBAG__CARD @ A_27a @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      = c_2Enum_2E0 ) ).

thf(thm_2Ebag_2EBCARD__0,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
     => ( ( ( c_2Ebag_2EBAG__CARD @ A_27a @ V0b )
          = c_2Enum_2E0 )
      <=> ( V0b
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).

thf(thm_2Ebag_2EBAG__CARD__THM,axiom,
    ! [A_27a: $tType] :
      ( ( ( c_2Ebag_2EBAG__CARD @ A_27a @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
        = c_2Enum_2E0 )
      & ! [V0b: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
         => ! [V1e: A_27a] :
              ( ( c_2Ebag_2EBAG__CARD @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1e @ V0b ) )
              = ( c_2Earithmetic_2E_2B @ ( c_2Ebag_2EBAG__CARD @ A_27a @ V0b ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__CARD__UNION,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b1 )
        & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b2 ) )
     => ( ( c_2Ebag_2EBAG__CARD @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) )
        = ( c_2Earithmetic_2E_2B @ ( c_2Ebag_2EBAG__CARD @ A_27a @ V0b1 ) @ ( c_2Ebag_2EBAG__CARD @ A_27a @ V1b2 ) ) ) ) ).

thf(thm_2Ebag_2EBCARD__SUC,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
     => ! [V1n: tyop_2Enum_2Enum] :
          ( ( ( c_2Ebag_2EBAG__CARD @ A_27a @ V0b )
            = ( c_2Enum_2ESUC @ V1n ) )
        <=> ? [V2b0: A_27a > tyop_2Enum_2Enum,V3e: A_27a] :
              ( ( V0b
                = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V3e @ V2b0 ) )
              & ( ( c_2Ebag_2EBAG__CARD @ A_27a @ V2b0 )
                = V1n ) ) ) ) ).

thf(thm_2Ebag_2EBAG__CARD__BAG__INN,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
     => ! [V1n: tyop_2Enum_2Enum,V2e: A_27a] :
          ( ( c_2Ebag_2EBAG__INN @ A_27a @ V2e @ V1n @ V0b )
         => ( c_2Earithmetic_2E_3C_3D @ V1n @ ( c_2Ebag_2EBAG__CARD @ A_27a @ V0b ) ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__DIFF__EQ,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 )
     => ( V1b2
        = ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1b2 @ V0b1 ) ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__CARD,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b2 )
        & ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 ) )
     => ( c_2Earithmetic_2E_3C_3D @ ( c_2Ebag_2EBAG__CARD @ A_27a @ V0b1 ) @ ( c_2Ebag_2EBAG__CARD @ A_27a @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__CARD__DIFF,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
     => ! [V1c: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V1c @ V0b )
         => ( ( c_2Ebag_2EBAG__CARD @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0b @ V1c ) )
            = ( c_2Earithmetic_2E_2D @ ( c_2Ebag_2EBAG__CARD @ A_27a @ V0b ) @ ( c_2Ebag_2EBAG__CARD @ A_27a @ V1c ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__FILTER__EMPTY,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o] :
      ( ( c_2Ebag_2EBAG__FILTER @ A_27a @ V0P @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ).

thf(thm_2Ebag_2EBAG__FILTER__BAG__INSERT,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b: A_27a > tyop_2Enum_2Enum,V2P: A_27a > $o] :
      ( ( c_2Ebag_2EBAG__FILTER @ A_27a @ V2P @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V1b ) )
      = ( c_2Ebool_2ECOND @ ( A_27a > tyop_2Enum_2Enum ) @ ( V2P @ V0e ) @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ ( c_2Ebag_2EBAG__FILTER @ A_27a @ V2P @ V1b ) ) @ ( c_2Ebag_2EBAG__FILTER @ A_27a @ V2P @ V1b ) ) ) ).

thf(thm_2Ebag_2EFINITE__BAG__FILTER,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b )
     => ( c_2Ebag_2EFINITE__BAG @ A_27a @ ( c_2Ebag_2EBAG__FILTER @ A_27a @ V0P @ V1b ) ) ) ).

thf(thm_2Ebag_2EBAG__INN__BAG__FILTER,axiom,
    ! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1e: A_27a,V2b: A_27a > tyop_2Enum_2Enum,V3P: A_27a > $o] :
      ( ( c_2Ebag_2EBAG__INN @ A_27a @ V1e @ V0n @ ( c_2Ebag_2EBAG__FILTER @ A_27a @ V3P @ V2b ) )
    <=> ( ( V0n = c_2Enum_2E0 )
        | ( ( V3P @ V1e )
          & ( c_2Ebag_2EBAG__INN @ A_27a @ V1e @ V0n @ V2b ) ) ) ) ).

thf(thm_2Ebag_2EBAG__IN__BAG__FILTER,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b: A_27a > tyop_2Enum_2Enum,V2P: A_27a > $o] :
      ( ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ ( c_2Ebag_2EBAG__FILTER @ A_27a @ V2P @ V1b ) )
    <=> ( ( V2P @ V0e )
        & ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V1b ) ) ) ).

thf(thm_2Ebag_2EBAG__FILTER__FILTER,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1Q: A_27a > $o,V2P: A_27a > $o] :
      ( ( c_2Ebag_2EBAG__FILTER @ A_27a @ V2P @ ( c_2Ebag_2EBAG__FILTER @ A_27a @ V1Q @ V0b ) )
      = ( c_2Ebag_2EBAG__FILTER @ A_27a
        @ ^ [V3a: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V2P @ V3a ) @ ( V1Q @ V3a ) )
        @ V0b ) ) ).

thf(thm_2Ebag_2EBAG__FILTER__SUB__BAG,axiom,
    ! [A_27a: $tType,V0P: A_27a > $o,V1b: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__FILTER @ A_27a @ V0P @ V1b ) @ V1b ) ).

thf(thm_2Ebag_2ESET__OF__BAG__EQ__INSERT,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1e: A_27a,V2s: A_27a > $o] :
      ( ( ( c_2Epred__set_2EINSERT @ A_27a @ V1e @ V2s )
        = ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b ) )
    <=> ? [V3b0: A_27a > tyop_2Enum_2Enum,V4eb: A_27a > tyop_2Enum_2Enum] :
          ( ( V0b
            = ( c_2Ebag_2EBAG__UNION @ A_27a @ V4eb @ V3b0 ) )
          & ( V2s
            = ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V3b0 ) )
          & ! [V5e_27: A_27a] :
              ( ( c_2Ebag_2EBAG__IN @ A_27a @ V5e_27 @ V4eb )
             => ( V5e_27 = V1e ) )
          & ( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1e @ V2s ) )
           => ( c_2Ebag_2EBAG__IN @ A_27a @ V1e @ V4eb ) ) ) ) ).

thf(thm_2Ebag_2EFINITE__SET__OF__BAG,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b ) )
      = ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b ) ) ).

thf(thm_2Ebag_2EBAG__IMAGE__EMPTY,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
      ( ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a @ V0f @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      = ( c_2Ebag_2EEMPTY__BAG @ A_27b ) ) ).

thf(thm_2Ebag_2EBAG__IMAGE__FINITE__INSERT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1f: A_27a > A_27b,V2e: A_27a] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
     => ( ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a @ V1f @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V2e @ V0b ) )
        = ( c_2Ebag_2EBAG__INSERT @ A_27b @ ( V1f @ V2e ) @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a @ V1f @ V0b ) ) ) ) ).

thf(thm_2Ebag_2EBAG__IMAGE__FINITE__UNION,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2f: A_27a > A_27b] :
      ( ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b1 )
        & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b2 ) )
     => ( ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a @ V2f @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) )
        = ( c_2Ebag_2EBAG__UNION @ A_27b @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a @ V2f @ V0b1 ) @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a @ V2f @ V1b2 ) ) ) ) ).

thf(thm_2Ebag_2EBAG__IMAGE__FINITE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b )
     => ( c_2Ebag_2EFINITE__BAG @ A_27b @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a @ V0f @ V1b ) ) ) ).

thf(thm_2Ebag_2EBAG__IMAGE__COMPOSE,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1g: A_27c > A_27a,V2b: A_27c > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27c @ V2b )
     => ( ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27c @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a @ V0f @ V1g ) @ V2b )
        = ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a @ V0f @ ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27c @ V1g @ V2b ) ) ) ) ).

thf(thm_2Ebag_2EBAG__IMAGE__FINITE__I,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
     => ( ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27a @ ( c_2Ecombin_2EI @ A_27a ) @ V0b )
        = V0b ) ) ).

thf(thm_2Ebag_2EBAG__IN__FINITE__BAG__IMAGE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0x: A_27b,V1f: A_27a > A_27b,V2b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V2b )
     => ( ( c_2Ebag_2EBAG__IN @ A_27b @ V0x @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a @ V1f @ V2b ) )
      <=> ? [V3y: A_27a] :
            ( ( ( V1f @ V3y )
              = V0x )
            & ( c_2Ebag_2EBAG__IN @ A_27a @ V3y @ V2b ) ) ) ) ).

thf(thm_2Ebag_2EBAG__IMAGE__EQ__EMPTY,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b )
     => ( ( ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a @ V0f @ V1b )
          = ( c_2Ebag_2EEMPTY__BAG @ A_27b ) )
      <=> ( V1b
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).

thf(thm_2Ebag_2EBAG__INSERT__CHOICE__REST,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( (~)
        @ ( V0b
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
     => ( V0b
        = ( c_2Ebag_2EBAG__INSERT @ A_27a @ ( c_2Ebag_2EBAG__CHOICE @ A_27a @ V0b ) @ ( c_2Ebag_2EBAG__REST @ A_27a @ V0b ) ) ) ) ).

thf(thm_2Ebag_2EBAG__CHOICE__SING,axiom,
    ! [A_27a: $tType,V0x: A_27a] :
      ( ( c_2Ebag_2EBAG__CHOICE @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
      = V0x ) ).

thf(thm_2Ebag_2EBAG__REST__SING,axiom,
    ! [A_27a: $tType,V0x: A_27a] :
      ( ( c_2Ebag_2EBAG__REST @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
      = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ).

thf(thm_2Ebag_2ESUB__BAG__REST,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__REST @ A_27a @ V0b ) @ V0b ) ).

thf(thm_2Ebag_2EPSUB__BAG__REST,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( (~)
        @ ( V0b
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
     => ( c_2Ebag_2EPSUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__REST @ A_27a @ V0b ) @ V0b ) ) ).

thf(thm_2Ebag_2ESUB__BAG__UNION__MONO,axiom,
    ! [A_27a: $tType] :
      ( ! [V0x: A_27a > tyop_2Enum_2Enum,V1y: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ V0x @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0x @ V1y ) )
      & ! [V2x: A_27a > tyop_2Enum_2Enum,V3y: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ V2x @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V3y @ V2x ) ) ) ).

thf(thm_2Ebag_2EPSUB__BAG__CARD,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b2 )
        & ( c_2Ebag_2EPSUB__BAG @ A_27a @ V0b1 @ V1b2 ) )
     => ( c_2Eprim__rec_2E_3C @ ( c_2Ebag_2EBAG__CARD @ A_27a @ V0b1 ) @ ( c_2Ebag_2EBAG__CARD @ A_27a @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EEL__BAG__BAG__INSERT,axiom,
    ! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2b: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1x @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
        = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0y @ V2b ) )
    <=> ( ( V1x = V0y )
        & ( V2b
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).

thf(thm_2Ebag_2EEL__BAG__SUB__BAG,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) @ V1b )
      = ( c_2Ebag_2EBAG__IN @ A_27a @ V0x @ V1b ) ) ).

thf(thm_2Ebag_2EITBAG__IND,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1P: ( A_27a > tyop_2Enum_2Enum ) > A_27b > $o] :
      ( ! [V2b: A_27a > tyop_2Enum_2Enum,V3acc: A_27b] :
          ( ( ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V2b )
              & ( (~)
                @ ( V2b
                  = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) )
           => ( V1P @ ( c_2Ebag_2EBAG__REST @ A_27a @ V2b ) @ ( V0f @ ( c_2Ebag_2EBAG__CHOICE @ A_27a @ V2b ) @ V3acc ) ) )
         => ( V1P @ V2b @ V3acc ) )
     => ! [V4v: A_27a > tyop_2Enum_2Enum,V5v1: A_27b] : ( V1P @ V4v @ V5v1 ) ) ).

thf(thm_2Ebag_2EITBAG__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1f: A_27a > A_27b > A_27b,V2acc: A_27b] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b )
     => ( ( c_2Ebag_2EITBAG @ A_27a @ A_27b @ V1f @ V0b @ V2acc )
        = ( c_2Ebool_2ECOND @ A_27b @ ( c_2Emin_2E_3D @ ( A_27a > tyop_2Enum_2Enum ) @ V0b @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) @ V2acc @ ( c_2Ebag_2EITBAG @ A_27a @ A_27b @ V1f @ ( c_2Ebag_2EBAG__REST @ A_27a @ V0b ) @ ( V1f @ ( c_2Ebag_2EBAG__CHOICE @ A_27a @ V0b ) @ V2acc ) ) ) ) ) ).

thf(thm_2Ebag_2EITBAG__EMPTY,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1acc: A_27b] :
      ( ( c_2Ebag_2EITBAG @ A_27a @ A_27b @ V0f @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) @ V1acc )
      = V1acc ) ).

thf(thm_2Ebag_2EITBAG__INSERT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1b: A_27a > tyop_2Enum_2Enum,V2f: A_27a > A_27b > A_27b,V3acc: A_27b] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b )
     => ( ( c_2Ebag_2EITBAG @ A_27a @ A_27b @ V2f @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ V1b ) @ V3acc )
        = ( c_2Ebag_2EITBAG @ A_27a @ A_27b @ V2f @ ( c_2Ebag_2EBAG__REST @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ V1b ) ) @ ( V2f @ ( c_2Ebag_2EBAG__CHOICE @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0x @ V1b ) ) @ V3acc ) ) ) ) ).

thf(thm_2Ebag_2ECOMMUTING__ITBAG__INSERT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( ! [V2x: A_27a,V3y: A_27a,V4z: A_27b] :
            ( ( V0f @ V2x @ ( V0f @ V3y @ V4z ) )
            = ( V0f @ V3y @ ( V0f @ V2x @ V4z ) ) )
        & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b ) )
     => ! [V5x: A_27a,V6a: A_27b] :
          ( ( c_2Ebag_2EITBAG @ A_27a @ A_27b @ V0f @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V5x @ V1b ) @ V6a )
          = ( c_2Ebag_2EITBAG @ A_27a @ A_27b @ V0f @ V1b @ ( V0f @ V5x @ V6a ) ) ) ) ).

thf(thm_2Ebag_2ECOMMUTING__ITBAG__RECURSES,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1e: A_27a,V2b: A_27a > tyop_2Enum_2Enum,V3a: A_27b] :
      ( ( ! [V4x: A_27a,V5y: A_27a,V6z: A_27b] :
            ( ( V0f @ V4x @ ( V0f @ V5y @ V6z ) )
            = ( V0f @ V5y @ ( V0f @ V4x @ V6z ) ) )
        & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V2b ) )
     => ( ( c_2Ebag_2EITBAG @ A_27a @ A_27b @ V0f @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1e @ V2b ) @ V3a )
        = ( V0f @ V1e @ ( c_2Ebag_2EITBAG @ A_27a @ A_27b @ V0f @ V2b @ V3a ) ) ) ) ).

thf(thm_2Ebag_2EBAG__GEN__SUM__EMPTY,axiom,
    ! [V0n: tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__GEN__SUM @ ( c_2Ebag_2EEMPTY__BAG @ tyop_2Enum_2Enum ) @ V0n )
      = V0n ) ).

thf(thm_2Ebag_2EBAG__GEN__PROD__EMPTY,axiom,
    ! [V0n: tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__GEN__PROD @ ( c_2Ebag_2EEMPTY__BAG @ tyop_2Enum_2Enum ) @ V0n )
      = V0n ) ).

thf(thm_2Ebag_2EBAG__GEN__SUM__TAILREC,axiom,
    ! [V0b: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V0b )
     => ! [V1x: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__GEN__SUM @ ( c_2Ebag_2EBAG__INSERT @ tyop_2Enum_2Enum @ V1x @ V0b ) @ V2a )
          = ( c_2Ebag_2EBAG__GEN__SUM @ V0b @ ( c_2Earithmetic_2E_2B @ V1x @ V2a ) ) ) ) ).

thf(thm_2Ebag_2EBAG__GEN__SUM__REC,axiom,
    ! [V0b: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V0b )
     => ! [V1x: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__GEN__SUM @ ( c_2Ebag_2EBAG__INSERT @ tyop_2Enum_2Enum @ V1x @ V0b ) @ V2a )
          = ( c_2Earithmetic_2E_2B @ V1x @ ( c_2Ebag_2EBAG__GEN__SUM @ V0b @ V2a ) ) ) ) ).

thf(thm_2Ebag_2EBAG__GEN__PROD__TAILREC,axiom,
    ! [V0b: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V0b )
     => ! [V1x: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__GEN__PROD @ ( c_2Ebag_2EBAG__INSERT @ tyop_2Enum_2Enum @ V1x @ V0b ) @ V2a )
          = ( c_2Ebag_2EBAG__GEN__PROD @ V0b @ ( c_2Earithmetic_2E_2A @ V1x @ V2a ) ) ) ) ).

thf(thm_2Ebag_2EBAG__GEN__PROD__REC,axiom,
    ! [V0b: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V0b )
     => ! [V1x: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__GEN__PROD @ ( c_2Ebag_2EBAG__INSERT @ tyop_2Enum_2Enum @ V1x @ V0b ) @ V2a )
          = ( c_2Earithmetic_2E_2A @ V1x @ ( c_2Ebag_2EBAG__GEN__PROD @ V0b @ V2a ) ) ) ) ).

thf(thm_2Ebag_2EBAG__GEN__PROD__EQ__1,axiom,
    ! [V0b: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V0b )
     => ! [V1e: tyop_2Enum_2Enum] :
          ( ( ( c_2Ebag_2EBAG__GEN__PROD @ V0b @ V1e )
            = ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
         => ( V1e
            = ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__GEN__PROD__ALL__ONES,axiom,
    ! [V0b: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V0b )
     => ( ( ( c_2Ebag_2EBAG__GEN__PROD @ V0b @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
          = ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
       => ! [V1x: tyop_2Enum_2Enum] :
            ( ( c_2Ebag_2EBAG__IN @ tyop_2Enum_2Enum @ V1x @ V0b )
           => ( V1x
              = ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__GEN__PROD__POSITIVE,axiom,
    ! [V0b: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V0b )
     => ( ! [V1x: tyop_2Enum_2Enum] :
            ( ( c_2Ebag_2EBAG__IN @ tyop_2Enum_2Enum @ V1x @ V0b )
           => ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1x ) )
       => ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Ebag_2EBAG__GEN__PROD @ V0b @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__EVERY__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ! [V0P: A_27a > $o] : ( c_2Ebag_2EBAG__EVERY @ A_27a @ V0P @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      & ! [V1P: A_27b > $o,V2e: A_27b,V3b: A_27b > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__EVERY @ A_27b @ V1P @ ( c_2Ebag_2EBAG__INSERT @ A_27b @ V2e @ V3b ) )
        <=> ( ( V1P @ V2e )
            & ( c_2Ebag_2EBAG__EVERY @ A_27b @ V1P @ V3b ) ) ) ) ).

thf(thm_2Ebag_2EBAG__EVERY__UNION,axiom,
    ! [A_27a: $tType,V0b2: A_27a > tyop_2Enum_2Enum,V1b1: A_27a > tyop_2Enum_2Enum,V2P: A_27a > $o] :
      ( ( c_2Ebag_2EBAG__EVERY @ A_27a @ V2P @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b1 @ V0b2 ) )
    <=> ( ( c_2Ebag_2EBAG__EVERY @ A_27a @ V2P @ V1b1 )
        & ( c_2Ebag_2EBAG__EVERY @ A_27a @ V2P @ V0b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__EVERY__MERGE,axiom,
    ! [A_27a: $tType,V0b2: A_27a > tyop_2Enum_2Enum,V1b1: A_27a > tyop_2Enum_2Enum,V2P: A_27a > $o] :
      ( ( c_2Ebag_2EBAG__EVERY @ A_27a @ V2P @ ( c_2Ebag_2EBAG__MERGE @ A_27a @ V1b1 @ V0b2 ) )
    <=> ( ( c_2Ebag_2EBAG__EVERY @ A_27a @ V2P @ V1b1 )
        & ( c_2Ebag_2EBAG__EVERY @ A_27a @ V2P @ V0b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__EVERY__SET,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1P: A_27a > $o] :
      ( ( c_2Ebag_2EBAG__EVERY @ A_27a @ V1P @ V0b )
      = ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b )
        @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
          @ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( V1P @ V2x ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__FILTER__EQ__EMPTY,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1P: A_27a > $o] :
      ( ( ( c_2Ebag_2EBAG__FILTER @ A_27a @ V1P @ V0b )
        = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
    <=> ( c_2Ebag_2EBAG__EVERY @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ $o @ $o @ c_2Ebool_2E_7E @ V1P ) @ V0b ) ) ).

thf(thm_2Ebag_2ESET__OF__BAG__IMAGE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27b > A_27a,V1b: A_27b > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESET__OF__BAG @ A_27a @ ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27b @ V0f @ V1b ) )
      = ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ V0f @ ( c_2Ebag_2ESET__OF__BAG @ A_27b @ V1b ) ) ) ).

thf(thm_2Ebag_2EBAG__IMAGE__FINITE__RESTRICTED__I,axiom,
    ! [A_27a: $tType,V0f: A_27a > A_27a,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b )
        & ( c_2Ebag_2EBAG__EVERY @ A_27a
          @ ^ [V2e: A_27a] : ( c_2Emin_2E_3D @ A_27a @ ( V0f @ V2e ) @ V2e )
          @ V1b ) )
     => ( ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27a @ V0f @ V1b )
        = V1b ) ) ).

thf(thm_2Ebag_2EBAG__ALL__DISTINCT__THM,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      & ! [V0e: A_27b,V1b: A_27b > tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27b @ ( c_2Ebag_2EBAG__INSERT @ A_27b @ V0e @ V1b ) )
        <=> ( ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27b @ V0e @ V1b ) )
            & ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27b @ V1b ) ) ) ) ).

thf(thm_2Ebag_2EBAG__ALL__DISTINCT__BAG__MERGE,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ ( c_2Ebag_2EBAG__MERGE @ A_27a @ V0b1 @ V1b2 ) )
    <=> ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V0b1 )
        & ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__ALL__DISTINCT__BAG__UNION,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b1 @ V1b2 ) )
    <=> ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V0b1 )
        & ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V1b2 )
        & ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ V0b1 @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__ALL__DISTINCT__DIFF,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V0b1 )
     => ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0b1 @ V1b2 ) ) ) ).

thf(thm_2Ebag_2EBAG__ALL__DISTINCT__DELETE,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V0b )
    <=> ! [V1e: A_27a] :
          ( ( c_2Ebag_2EBAG__IN @ A_27a @ V1e @ V0b )
         => ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27a @ V1e @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0b @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1e @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__ALL__DISTINCT__SET,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V0b )
    <=> ( ( c_2Ebag_2EBAG__OF__SET @ A_27a @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0b ) )
        = V0b ) ) ).

thf(thm_2Ebag_2EBAG__ALL__DISTINCT__BAG__OF__SET,axiom,
    ! [A_27a: $tType,V0s: A_27a > $o] : ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ ( c_2Ebag_2EBAG__OF__SET @ A_27a @ V0s ) ) ).

thf(thm_2Ebag_2EBAG__IN__BAG__DIFF__ALL__DISTINCT,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum,V2e: A_27a] :
      ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V0b1 )
     => ( ( c_2Ebag_2EBAG__IN @ A_27a @ V2e @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0b1 @ V1b2 ) )
      <=> ( ( c_2Ebag_2EBAG__IN @ A_27a @ V2e @ V0b1 )
          & ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27a @ V2e @ V1b2 ) ) ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__ALL__DISTINCT,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V0b1 )
     => ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V0b1 @ V1b2 )
      <=> ! [V2x: A_27a] :
            ( ( c_2Ebag_2EBAG__IN @ A_27a @ V2x @ V0b1 )
           => ( c_2Ebag_2EBAG__IN @ A_27a @ V2x @ V1b2 ) ) ) ) ).

thf(thm_2Ebag_2EBAG__ALL__DISTINCT__BAG__INN,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2e: A_27a] :
      ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V0b )
     => ( ( c_2Ebag_2EBAG__INN @ A_27a @ V2e @ V1n @ V0b )
      <=> ( ( V1n = c_2Enum_2E0 )
          | ( ( V1n
              = ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
            & ( c_2Ebag_2EBAG__IN @ A_27a @ V2e @ V0b ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__ALL__DISTINCT__EXTENSION,axiom,
    ! [A_27a: $tType,V0b1: A_27a > tyop_2Enum_2Enum,V1b2: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V0b1 )
        & ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a @ V1b2 ) )
     => ( ( V0b1 = V1b2 )
      <=> ! [V2x: A_27a] :
            ( ( c_2Ebag_2EBAG__IN @ A_27a @ V2x @ V0b1 )
            = ( c_2Ebag_2EBAG__IN @ A_27a @ V2x @ V1b2 ) ) ) ) ).

thf(thm_2Ebag_2ENOT__BAG__IN,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1x: A_27a] :
      ( ( ( V0b @ V1x )
        = c_2Enum_2E0 )
    <=> ( (~) @ ( c_2Ebag_2EBAG__IN @ A_27a @ V1x @ V0b ) ) ) ).

thf(thm_2Ebag_2EBAG__UNION__EQ__LEFT,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1c: A_27a > tyop_2Enum_2Enum,V2d: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b @ V1c )
        = ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b @ V2d ) )
     => ( V1c = V2d ) ) ).

thf(thm_2Ebag_2EBAG__IN__DIVIDES,axiom,
    ! [V0b: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1x: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V0b )
        & ( c_2Ebag_2EBAG__IN @ tyop_2Enum_2Enum @ V1x @ V0b ) )
     => ( c_2Edivides_2Edivides @ V1x @ ( c_2Ebag_2EBAG__GEN__PROD @ V0b @ V2a ) ) ) ).

thf(thm_2Ebag_2EBAG__GEN__PROD__UNION__LEM,axiom,
    ! [V0b1: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V0b1 )
     => ! [V1b2: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum,V3b: tyop_2Enum_2Enum] :
          ( ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V1b2 )
         => ( ( c_2Ebag_2EBAG__GEN__PROD @ ( c_2Ebag_2EBAG__UNION @ tyop_2Enum_2Enum @ V0b1 @ V1b2 ) @ ( c_2Earithmetic_2E_2A @ V2a @ V3b ) )
            = ( c_2Earithmetic_2E_2A @ ( c_2Ebag_2EBAG__GEN__PROD @ V0b1 @ V2a ) @ ( c_2Ebag_2EBAG__GEN__PROD @ V1b2 @ V3b ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__GEN__PROD__UNION,axiom,
    ! [V0b1: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1b2: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V0b1 )
        & ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V1b2 ) )
     => ( ( c_2Ebag_2EBAG__GEN__PROD @ ( c_2Ebag_2EBAG__UNION @ tyop_2Enum_2Enum @ V0b1 @ V1b2 ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
        = ( c_2Earithmetic_2E_2A @ ( c_2Ebag_2EBAG__GEN__PROD @ V0b1 @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ebag_2EBAG__GEN__PROD @ V1b2 @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).

thf(thm_2Ebag_2EBIG__BAG__UNION__EMPTY,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ ( c_2Epred__set_2EEMPTY @ ( A_27a > tyop_2Enum_2Enum ) ) )
      = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ).

thf(thm_2Ebag_2EBIG__BAG__UNION__INSERT,axiom,
    ! [A_27a: $tType,V0sob: ( A_27a > tyop_2Enum_2Enum ) > $o,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Epred__set_2EFINITE @ ( A_27a > tyop_2Enum_2Enum ) @ V0sob )
     => ( ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ ( c_2Epred__set_2EINSERT @ ( A_27a > tyop_2Enum_2Enum ) @ V1b @ V0sob ) )
        = ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ ( c_2Epred__set_2EDELETE @ ( A_27a > tyop_2Enum_2Enum ) @ V0sob @ V1b ) ) ) ) ) ).

thf(thm_2Ebag_2EBIG__BAG__UNION__DELETE,axiom,
    ! [A_27a: $tType,V0sob: ( A_27a > tyop_2Enum_2Enum ) > $o,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Epred__set_2EFINITE @ ( A_27a > tyop_2Enum_2Enum ) @ V0sob )
     => ( ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ ( c_2Epred__set_2EDELETE @ ( A_27a > tyop_2Enum_2Enum ) @ V0sob @ V1b ) )
        = ( c_2Ebool_2ECOND @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebool_2EIN @ ( A_27a > tyop_2Enum_2Enum ) @ V1b @ V0sob ) @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ V0sob ) @ V1b ) @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ V0sob ) ) ) ) ).

thf(thm_2Ebag_2EBIG__BAG__UNION__ITSET__BAG__UNION,axiom,
    ! [A_27a: $tType,V0sob: ( A_27a > tyop_2Enum_2Enum ) > $o] :
      ( ( c_2Epred__set_2EFINITE @ ( A_27a > tyop_2Enum_2Enum ) @ V0sob )
     => ( ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ V0sob )
        = ( c_2Epred__set_2EITSET @ ( A_27a > tyop_2Enum_2Enum ) @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0sob @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).

thf(thm_2Ebag_2EFINITE__BIG__BAG__UNION,axiom,
    ! [A_27a: $tType,V0sob: ( A_27a > tyop_2Enum_2Enum ) > $o] :
      ( ( ( c_2Epred__set_2EFINITE @ ( A_27a > tyop_2Enum_2Enum ) @ V0sob )
        & ! [V1b: A_27a > tyop_2Enum_2Enum] :
            ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Enum_2Enum ) @ V1b @ V0sob )
           => ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b ) ) )
     => ( c_2Ebag_2EFINITE__BAG @ A_27a @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ V0sob ) ) ) ).

thf(thm_2Ebag_2EBAG__IN__BIG__BAG__UNION,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1P: ( A_27a > tyop_2Enum_2Enum ) > $o] :
      ( ( c_2Epred__set_2EFINITE @ ( A_27a > tyop_2Enum_2Enum ) @ V1P )
     => ( ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ V1P ) )
      <=> ? [V2b: A_27a > tyop_2Enum_2Enum] :
            ( ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V2b )
            & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Enum_2Enum ) @ V2b @ V1P ) ) ) ) ).

thf(thm_2Ebag_2EBIG__BAG__UNION__EQ__EMPTY__BAG,axiom,
    ! [A_27a: $tType,V0sob: ( A_27a > tyop_2Enum_2Enum ) > $o] :
      ( ( c_2Epred__set_2EFINITE @ ( A_27a > tyop_2Enum_2Enum ) @ V0sob )
     => ( ( ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ V0sob )
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      <=> ! [V1b: A_27a > tyop_2Enum_2Enum] :
            ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Enum_2Enum ) @ V1b @ V0sob )
           => ( V1b
              = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ).

thf(thm_2Ebag_2EBIG__BAG__UNION__UNION,axiom,
    ! [A_27a: $tType,V0s2: ( A_27a > tyop_2Enum_2Enum ) > $o,V1s1: ( A_27a > tyop_2Enum_2Enum ) > $o] :
      ( ( ( c_2Epred__set_2EFINITE @ ( A_27a > tyop_2Enum_2Enum ) @ V1s1 )
        & ( c_2Epred__set_2EFINITE @ ( A_27a > tyop_2Enum_2Enum ) @ V0s2 ) )
     => ( ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ ( c_2Epred__set_2EUNION @ ( A_27a > tyop_2Enum_2Enum ) @ V1s1 @ V0s2 ) )
        = ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ V1s1 ) @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ V0s2 ) ) @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ ( c_2Epred__set_2EINTER @ ( A_27a > tyop_2Enum_2Enum ) @ V1s1 @ V0s2 ) ) ) ) ) ).

thf(thm_2Ebag_2EBIG__BAG__UNION__EQ__ELEMENT,axiom,
    ! [A_27a: $tType,V0sob: ( A_27a > tyop_2Enum_2Enum ) > $o,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Epred__set_2EFINITE @ ( A_27a > tyop_2Enum_2Enum ) @ V0sob )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Enum_2Enum ) @ V1b @ V0sob ) )
     => ( ( ( c_2Ebag_2EBIG__BAG__UNION @ A_27a @ V0sob )
          = V1b )
      <=> ! [V2b_27: A_27a > tyop_2Enum_2Enum] :
            ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Enum_2Enum ) @ V2b_27 @ V0sob )
           => ( ( V2b_27 = V1b )
              | ( V2b_27
                = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__NOT__LESS__EMPTY,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o,V1b: A_27a > tyop_2Enum_2Enum] : ( (~) @ ( c_2Ebag_2Emlt1 @ A_27a @ V0r @ V1b @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ).

thf(thm_2Ebag_2ENOT__mlt__EMPTY,axiom,
    ! [A_27a: $tType,V0b: A_27a > tyop_2Enum_2Enum,V1R: A_27a > A_27a > $o] : ( (~) @ ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V1R ) @ V0b @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ).

thf(thm_2Ebag_2EBAG__LESS__ADD,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o,V1a: A_27a,V2N: A_27a > tyop_2Enum_2Enum,V3M0: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2Emlt1 @ A_27a @ V0r @ V2N @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V3M0 @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1a @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) )
     => ( ? [V4M: A_27a > tyop_2Enum_2Enum] :
            ( ( c_2Ebag_2Emlt1 @ A_27a @ V0r @ V4M @ V3M0 )
            & ( V2N
              = ( c_2Ebag_2EBAG__UNION @ A_27a @ V4M @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1a @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) )
        | ? [V5KK: A_27a > tyop_2Enum_2Enum] :
            ( ! [V6b: A_27a] :
                ( ( c_2Ebag_2EBAG__IN @ A_27a @ V6b @ V5KK )
               => ( V0r @ V6b @ V1a ) )
            & ( V2N
              = ( c_2Ebag_2EBAG__UNION @ A_27a @ V3M0 @ V5KK ) ) ) ) ) ).

thf(thm_2Ebag_2Emlt1__all__accessible,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EWF @ A_27a @ V0R )
     => ! [V1M: A_27a > tyop_2Enum_2Enum] : ( c_2Erelation_2EWFP @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V0R ) @ V1M ) ) ).

thf(thm_2Ebag_2EWF__mlt1,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EWF @ A_27a @ V0R )
     => ( c_2Erelation_2EWF @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V0R ) ) ) ).

thf(thm_2Ebag_2ETC__mlt1__FINITE__BAG,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V0R ) @ V1b1 @ V2b2 )
     => ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b1 )
        & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V2b2 ) ) ) ).

thf(thm_2Ebag_2ETC__mlt1__UNION2__I,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1b2: A_27a > tyop_2Enum_2Enum,V2b1: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b2 )
        & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V2b1 )
        & ( (~)
          @ ( V1b2
            = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) )
     => ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V0R ) @ V2b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V2b1 @ V1b2 ) ) ) ).

thf(thm_2Ebag_2ETC__mlt1__UNION1__I,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1b2: A_27a > tyop_2Enum_2Enum,V2b1: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b2 )
        & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V2b1 )
        & ( (~)
          @ ( V2b1
            = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) )
     => ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V0R ) @ V1b2 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V2b1 @ V1b2 ) ) ) ).

thf(thm_2Ebag_2Emlt__NOT__REFL,axiom,
    ! [A_27a: $tType,V0a: A_27a > tyop_2Enum_2Enum,V1R: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EWF @ A_27a @ V1R )
     => ( (~) @ ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V1R ) @ V0a @ V0a ) ) ) ).

thf(thm_2Ebag_2Emlt__INSERT__CANCEL__I,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1R: A_27a > A_27a > $o,V2a: A_27a > tyop_2Enum_2Enum,V3b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V1R ) @ V2a @ V3b )
     => ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V1R ) @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V2a ) @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V3b ) ) ) ).

thf(thm_2Ebag_2Emlt1__INSERT__CANCEL,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b: A_27a > tyop_2Enum_2Enum,V2a: A_27a > tyop_2Enum_2Enum,V3R: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EWF @ A_27a @ V3R )
     => ( ( c_2Ebag_2Emlt1 @ A_27a @ V3R @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V2a ) @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V1b ) )
        = ( c_2Ebag_2Emlt1 @ A_27a @ V3R @ V2a @ V1b ) ) ) ).

thf(thm_2Ebag_2Edominates__EMPTY,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0b: A_27b > $o,V1R: A_27a > A_27b > $o] : ( c_2Ebag_2Edominates @ A_27a @ A_27b @ V1R @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0b ) ).

thf(thm_2Ebag_2Edominates__SUBSET,axiom,
    ! [A_27a: $tType,V0Y: A_27a > $o,V1X: A_27a > $o,V2R: A_27a > A_27a > $o] :
      ( ( ( c_2Erelation_2Etransitive @ A_27a @ V2R )
        & ( c_2Epred__set_2EFINITE @ A_27a @ V0Y )
        & ( c_2Ebag_2Edominates @ A_27a @ A_27a @ V2R @ V0Y @ V1X )
        & ( c_2Epred__set_2ESUBSET @ A_27a @ V1X @ V0Y )
        & ( (~)
          @ ( V1X
            = ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
     => ? [V3x: A_27a] :
          ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1X )
          & ( V2R @ V3x @ V3x ) ) ) ).

thf(thm_2Ebag_2Emlt__dominates__thm1,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2Etransitive @ A_27a @ V0R )
     => ! [V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V0R ) @ V1b1 @ V2b2 )
        <=> ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b1 )
            & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V2b2 )
            & ? [V3x: A_27a > tyop_2Enum_2Enum,V4y: A_27a > tyop_2Enum_2Enum] :
                ( ( (~)
                  @ ( V3x
                    = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
                & ( c_2Ebag_2ESUB__BAG @ A_27a @ V3x @ V2b2 )
                & ( V1b1
                  = ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V2b2 @ V3x ) @ V4y ) )
                & ( c_2Ebag_2Edominates @ A_27a @ A_27a @ V0R @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V4y ) @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V3x ) ) ) ) ) ) ).

thf(thm_2Ebag_2Edominates__DIFF,axiom,
    ! [A_27a: $tType,V0y: A_27a > $o,V1x: A_27a > $o,V2i: A_27a > $o,V3R: A_27a > A_27a > $o] :
      ( ( ( c_2Erelation_2EWF @ A_27a @ V3R )
        & ( c_2Erelation_2Etransitive @ A_27a @ V3R )
        & ( c_2Ebag_2Edominates @ A_27a @ A_27a @ V3R @ V1x @ V0y )
        & ( c_2Epred__set_2EFINITE @ A_27a @ V2i )
        & ( c_2Epred__set_2ESUBSET @ A_27a @ V2i @ V1x )
        & ( c_2Epred__set_2ESUBSET @ A_27a @ V2i @ V0y ) )
     => ( c_2Ebag_2Edominates @ A_27a @ A_27a @ V3R @ ( c_2Epred__set_2EDIFF @ A_27a @ V1x @ V2i ) @ ( c_2Epred__set_2EDIFF @ A_27a @ V0y @ V2i ) ) ) ).

thf(thm_2Ebag_2EBAG__INSERT__SUB__BAG__E,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1c: A_27a > tyop_2Enum_2Enum,V2b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V2b ) @ V1c )
     => ( ( c_2Ebag_2EBAG__IN @ A_27a @ V0e @ V1c )
        & ( c_2Ebag_2ESUB__BAG @ A_27a @ V2b @ V1c ) ) ) ).

thf(thm_2Ebag_2Ebdominates__BAG__DIFF,axiom,
    ! [A_27a: $tType,V0y: A_27a > tyop_2Enum_2Enum,V1x: A_27a > tyop_2Enum_2Enum,V2i: A_27a > tyop_2Enum_2Enum,V3R: A_27a > A_27a > $o] :
      ( ( ( c_2Erelation_2EWF @ A_27a @ V3R )
        & ( c_2Erelation_2Etransitive @ A_27a @ V3R )
        & ( c_2Ebag_2Edominates @ A_27a @ A_27a @ V3R @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V1x ) @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V0y ) )
        & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V2i )
        & ( c_2Ebag_2ESUB__BAG @ A_27a @ V2i @ V1x )
        & ( c_2Ebag_2ESUB__BAG @ A_27a @ V2i @ V0y ) )
     => ( c_2Ebag_2Edominates @ A_27a @ A_27a @ V3R @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1x @ V2i ) ) @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V0y @ V2i ) ) ) ) ).

thf(thm_2Ebag_2EBAG__INTER__SUB__BAG,axiom,
    ! [A_27a: $tType,V0b2: A_27a > tyop_2Enum_2Enum,V1b1: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__INTER @ A_27a @ V1b1 @ V0b2 ) @ V1b1 )
      & ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__INTER @ A_27a @ V1b1 @ V0b2 ) @ V0b2 ) ) ).

thf(thm_2Ebag_2EBAG__INTER__FINITE,axiom,
    ! [A_27a: $tType,V0b2: A_27a > tyop_2Enum_2Enum,V1b1: A_27a > tyop_2Enum_2Enum] :
      ( ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b1 )
        | ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b2 ) )
     => ( c_2Ebag_2EFINITE__BAG @ A_27a @ ( c_2Ebag_2EBAG__INTER @ A_27a @ V1b1 @ V0b2 ) ) ) ).

thf(thm_2Ebag_2Emlt__dominates__thm2,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( ( c_2Erelation_2EWF @ A_27a @ V0R )
        & ( c_2Erelation_2Etransitive @ A_27a @ V0R ) )
     => ! [V1b1: A_27a > tyop_2Enum_2Enum,V2b2: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V0R ) @ V1b1 @ V2b2 )
        <=> ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b1 )
            & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V2b2 )
            & ? [V3x: A_27a > tyop_2Enum_2Enum,V4y: A_27a > tyop_2Enum_2Enum] :
                ( ( (~)
                  @ ( V3x
                    = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
                & ( c_2Ebag_2ESUB__BAG @ A_27a @ V3x @ V2b2 )
                & ( c_2Ebag_2EBAG__DISJOINT @ A_27a @ V3x @ V4y )
                & ( V1b1
                  = ( c_2Ebag_2EBAG__UNION @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V2b2 @ V3x ) @ V4y ) )
                & ( c_2Ebag_2Edominates @ A_27a @ A_27a @ V0R @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V4y ) @ ( c_2Ebag_2ESET__OF__BAG @ A_27a @ V3x ) ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__DIFF__INSERT__SUB__BAG,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1c: A_27a > tyop_2Enum_2Enum,V2b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V1c @ V2b )
     => ( ( c_2Ebag_2EBAG__DIFF @ A_27a @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V2b ) @ V1c )
        = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V2b @ V1c ) ) ) ) ).

thf(thm_2Ebag_2Emlt__INSERT__CANCEL,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b: A_27a > tyop_2Enum_2Enum,V2a: A_27a > tyop_2Enum_2Enum,V3R: A_27a > A_27a > $o] :
      ( ( ( c_2Erelation_2Etransitive @ A_27a @ V3R )
        & ( c_2Erelation_2EWF @ A_27a @ V3R ) )
     => ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V3R ) @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V2a ) @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ V1b ) )
        = ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V3R ) @ V2a @ V1b ) ) ) ).

thf(thm_2Ebag_2Emlt__UNION__RCANCEL__I,axiom,
    ! [A_27a: $tType,V0c: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum,V2a: A_27a > tyop_2Enum_2Enum,V3R: A_27a > A_27a > $o] :
      ( ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V3R ) @ V2a @ V1b )
        & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0c ) )
     => ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V3R ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V2a @ V0c ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b @ V0c ) ) ) ).

thf(thm_2Ebag_2Emlt__UNION__RCANCEL,axiom,
    ! [A_27a: $tType,V0c: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum,V2a: A_27a > tyop_2Enum_2Enum,V3R: A_27a > A_27a > $o] :
      ( ( ( c_2Erelation_2EWF @ A_27a @ V3R )
        & ( c_2Erelation_2Etransitive @ A_27a @ V3R ) )
     => ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V3R ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V2a @ V0c ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b @ V0c ) )
      <=> ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V3R ) @ V2a @ V1b )
          & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0c ) ) ) ) ).

thf(thm_2Ebag_2Emlt__UNION__LCANCEL__I,axiom,
    ! [A_27a: $tType,V0c: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum,V2a: A_27a > tyop_2Enum_2Enum,V3R: A_27a > A_27a > $o] :
      ( ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V3R ) @ V2a @ V1b )
        & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0c ) )
     => ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V3R ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0c @ V2a ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0c @ V1b ) ) ) ).

thf(thm_2Ebag_2Emlt__UNION__LCANCEL,axiom,
    ! [A_27a: $tType,V0c: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum,V2a: A_27a > tyop_2Enum_2Enum,V3R: A_27a > A_27a > $o] :
      ( ( ( c_2Erelation_2EWF @ A_27a @ V3R )
        & ( c_2Erelation_2Etransitive @ A_27a @ V3R ) )
     => ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V3R ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0c @ V2a ) @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0c @ V1b ) )
      <=> ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V3R ) @ V2a @ V1b )
          & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0c ) ) ) ) ).

thf(thm_2Ebag_2Emlt__UNION__CANCEL__EQN,axiom,
    ! [A_27a: $tType,V0b2: A_27a > tyop_2Enum_2Enum,V1b1: A_27a > tyop_2Enum_2Enum,V2R: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EWF @ A_27a @ V2R )
     => ( ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V2R ) @ V1b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V1b1 @ V0b2 ) )
        <=> ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b1 )
            & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b2 )
            & ( (~)
              @ ( V0b2
                = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) )
        & ( ( c_2Erelation_2ETC @ ( A_27a > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ A_27a @ V2R ) @ V1b1 @ ( c_2Ebag_2EBAG__UNION @ A_27a @ V0b2 @ V1b1 ) )
        <=> ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V1b1 )
            & ( c_2Ebag_2EFINITE__BAG @ A_27a @ V0b2 )
            & ( (~)
              @ ( V0b2
                = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ) ).

thf(thm_2Ebag_2Emlt__UNION__EMPTY__EQN,axiom,
    c_2Ebool_2ET ).

thf(thm_2Ebag_2ESUB__BAG__SING,axiom,
    ! [A_27a: $tType,V0e: A_27a,V1b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2ESUB__BAG @ A_27a @ V1b @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
    <=> ( ( V1b
          = ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
        | ( V1b
          = ( c_2Ebag_2EBAG__INSERT @ A_27a @ V0e @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ).

thf(thm_2Ebag_2ESUB__BAG__DIFF__simple,axiom,
    ! [A_27a: $tType,V0c: A_27a > tyop_2Enum_2Enum,V1b: A_27a > tyop_2Enum_2Enum] : ( c_2Ebag_2ESUB__BAG @ A_27a @ ( c_2Ebag_2EBAG__DIFF @ A_27a @ V1b @ V0c ) @ V1b ) ).

thf(thm_2Ebag_2EmltLT__SING0,axiom,
    ! [V0b: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
      ( ( c_2Erelation_2ETC @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum ) @ ( c_2Ebag_2Emlt1 @ tyop_2Enum_2Enum @ c_2Eprim__rec_2E_3C ) @ ( c_2Ebag_2EBAG__INSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Ebag_2EEMPTY__BAG @ tyop_2Enum_2Enum ) ) @ V0b )
    <=> ( ( c_2Ebag_2EFINITE__BAG @ tyop_2Enum_2Enum @ V0b )
        & ( (~)
          @ ( V0b
            = ( c_2Ebag_2EBAG__INSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Ebag_2EEMPTY__BAG @ tyop_2Enum_2Enum ) ) ) )
        & ( (~)
          @ ( V0b
            = ( c_2Ebag_2EEMPTY__BAG @ tyop_2Enum_2Enum ) ) ) ) ) ).

thf(thm_2Ebag_2EBAG__SIZE__EMPTY,axiom,
    ! [A_27a: $tType,V0eltsize: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2Ebag__size @ A_27a @ V0eltsize @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
      = c_2Enum_2E0 ) ).

thf(thm_2Ebag_2EBAG__SIZE__INSERT,axiom,
    ! [A_27a: $tType,V0eltsize: A_27a > tyop_2Enum_2Enum,V1e: A_27a,V2b: A_27a > tyop_2Enum_2Enum] :
      ( ( c_2Ebag_2EFINITE__BAG @ A_27a @ V2b )
     => ( ( c_2Ebag_2Ebag__size @ A_27a @ V0eltsize @ ( c_2Ebag_2EBAG__INSERT @ A_27a @ V1e @ V2b ) )
        = ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( V0eltsize @ V1e ) ) @ ( c_2Ebag_2Ebag__size @ A_27a @ V0eltsize @ V2b ) ) ) ) ).

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