ITP001 Axioms: ITP045^5.ax
%------------------------------------------------------------------------------
% File : ITP045^5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : bag^2.ax [Gau20]
% : HL4045^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 324 ( 9 unt; 32 typ; 0 def)
% Number of atoms : 5795 ( 211 equ; 0 cnn)
% Maximal formula atoms : 692 ( 17 avg)
% Number of connectives : 10895 ( 42 ~; 18 |; 205 &;9614 @)
% ( 98 <=>; 918 =>; 0 <=; 0 <~>)
% Maximal formula depth : 47 ( 13 avg;9614 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 33 ( 33 >; 0 *; 0 +; 0 <<)
% Number of symbols : 85 ( 83 usr; 54 con; 0-2 aty)
% Number of variables : 1115 ( 20 ^1064 !; 31 ?;1115 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Ebag_2EBAG__ALL__DISTINCT,type,
c_2Ebag_2EBAG__ALL__DISTINCT: del > $i ).
thf(mem_c_2Ebag_2EBAG__ALL__DISTINCT,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) ) ).
thf(tp_c_2Ebag_2EBAG__CARD,type,
c_2Ebag_2EBAG__CARD: del > $i ).
thf(mem_c_2Ebag_2EBAG__CARD,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Ebag_2EBAG__CARD__RELn,type,
c_2Ebag_2EBAG__CARD__RELn: del > $i ).
thf(mem_c_2Ebag_2EBAG__CARD__RELn,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__CARD__RELn @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ).
thf(tp_c_2Ebag_2EBAG__CHOICE,type,
c_2Ebag_2EBAG__CHOICE: del > $i ).
thf(mem_c_2Ebag_2EBAG__CHOICE,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__CHOICE @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ A_27a ) ) ).
thf(tp_c_2Ebag_2EBAG__DELETE,type,
c_2Ebag_2EBAG__DELETE: del > $i ).
thf(mem_c_2Ebag_2EBAG__DELETE,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) ) ) ) ).
thf(tp_c_2Ebag_2EBAG__DIFF,type,
c_2Ebag_2EBAG__DIFF: del > $i ).
thf(mem_c_2Ebag_2EBAG__DIFF,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ) ).
thf(tp_c_2Ebag_2EBAG__DISJOINT,type,
c_2Ebag_2EBAG__DISJOINT: del > $i ).
thf(mem_c_2Ebag_2EBAG__DISJOINT,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) ) ) ).
thf(tp_c_2Ebag_2EBAG__EVERY,type,
c_2Ebag_2EBAG__EVERY: del > $i ).
thf(mem_c_2Ebag_2EBAG__EVERY,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__EVERY @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) ) ) ).
thf(tp_c_2Ebag_2EBAG__FILTER,type,
c_2Ebag_2EBAG__FILTER: del > $i ).
thf(mem_c_2Ebag_2EBAG__FILTER,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ) ).
thf(tp_c_2Ebag_2EBAG__GEN__PROD,type,
c_2Ebag_2EBAG__GEN__PROD: $i ).
thf(mem_c_2Ebag_2EBAG__GEN__PROD,axiom,
mem @ c_2Ebag_2EBAG__GEN__PROD @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Ebag_2EBAG__GEN__SUM,type,
c_2Ebag_2EBAG__GEN__SUM: $i ).
thf(mem_c_2Ebag_2EBAG__GEN__SUM,axiom,
mem @ c_2Ebag_2EBAG__GEN__SUM @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Ebag_2EBAG__IMAGE,type,
c_2Ebag_2EBAG__IMAGE: del > del > $i ).
thf(mem_c_2Ebag_2EBAG__IMAGE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ A_27a ) @ ( arr @ ( arr @ A_27b @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ) ).
thf(tp_c_2Ebag_2EBAG__IN,type,
c_2Ebag_2EBAG__IN: del > $i ).
thf(mem_c_2Ebag_2EBAG__IN,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) ) ) ).
thf(tp_c_2Ebag_2EBAG__INN,type,
c_2Ebag_2EBAG__INN: del > $i ).
thf(mem_c_2Ebag_2EBAG__INN,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) ) ) ) ).
thf(tp_c_2Ebag_2EBAG__INSERT,type,
c_2Ebag_2EBAG__INSERT: del > $i ).
thf(mem_c_2Ebag_2EBAG__INSERT,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ) ).
thf(tp_c_2Ebag_2EBAG__INTER,type,
c_2Ebag_2EBAG__INTER: del > $i ).
thf(mem_c_2Ebag_2EBAG__INTER,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__INTER @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ) ).
thf(tp_c_2Ebag_2EBAG__MERGE,type,
c_2Ebag_2EBAG__MERGE: del > $i ).
thf(mem_c_2Ebag_2EBAG__MERGE,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__MERGE @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ) ).
thf(tp_c_2Ebag_2EBAG__OF__SET,type,
c_2Ebag_2EBAG__OF__SET: del > $i ).
thf(mem_c_2Ebag_2EBAG__OF__SET,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__OF__SET @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Ebag_2EBAG__REST,type,
c_2Ebag_2EBAG__REST: del > $i ).
thf(mem_c_2Ebag_2EBAG__REST,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Ebag_2EBAG__UNION,type,
c_2Ebag_2EBAG__UNION: del > $i ).
thf(mem_c_2Ebag_2EBAG__UNION,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ) ).
thf(tp_c_2Ebag_2EBIG__BAG__UNION,type,
c_2Ebag_2EBIG__BAG__UNION: del > $i ).
thf(mem_c_2Ebag_2EBIG__BAG__UNION,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Ebag_2EEL__BAG,type,
c_2Ebag_2EEL__BAG: del > $i ).
thf(mem_c_2Ebag_2EEL__BAG,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EEL__BAG @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Ebag_2EEMPTY__BAG,type,
c_2Ebag_2EEMPTY__BAG: del > $i ).
thf(mem_c_2Ebag_2EEMPTY__BAG,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Ebag_2EFINITE__BAG,type,
c_2Ebag_2EFINITE__BAG: del > $i ).
thf(mem_c_2Ebag_2EFINITE__BAG,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) ) ).
thf(tp_c_2Ebag_2EITBAG,type,
c_2Ebag_2EITBAG: del > del > $i ).
thf(mem_c_2Ebag_2EITBAG,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ebag_2EITBAG @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27b @ A_27b ) ) ) ) ).
thf(tp_c_2Ebag_2EPSUB__BAG,type,
c_2Ebag_2EPSUB__BAG: del > $i ).
thf(mem_c_2Ebag_2EPSUB__BAG,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) ) ) ).
thf(tp_c_2Ebag_2ESET__OF__BAG,type,
c_2Ebag_2ESET__OF__BAG: del > $i ).
thf(mem_c_2Ebag_2ESET__OF__BAG,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Ebag_2ESING__BAG,type,
c_2Ebag_2ESING__BAG: del > $i ).
thf(mem_c_2Ebag_2ESING__BAG,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2ESING__BAG @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) ) ).
thf(tp_c_2Ebag_2ESUB__BAG,type,
c_2Ebag_2ESUB__BAG: del > $i ).
thf(mem_c_2Ebag_2ESUB__BAG,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) ) ) ).
thf(tp_c_2Ebag_2Ebag__size,type,
c_2Ebag_2Ebag__size: del > $i ).
thf(mem_c_2Ebag_2Ebag__size,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2Ebag__size @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Ebag_2Edominates,type,
c_2Ebag_2Edominates: del > del > $i ).
thf(mem_c_2Ebag_2Edominates,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ebag_2Edominates @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Ebag_2Emlt1,type,
c_2Ebag_2Emlt1: del > $i ).
thf(mem_c_2Ebag_2Emlt1,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebag_2Emlt1 @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) ) ) ) ).
thf(ax_thm_2Ebag_2EEMPTY__BAG,axiom,
! [A_27a: del] :
( ( c_2Ebag_2EEMPTY__BAG @ A_27a )
= ( ap @ ( c_2Ecombin_2EK @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Ebag_2EEMPTY__BAG__alt,axiom,
! [A_27a: del] :
( ( c_2Ebag_2EEMPTY__BAG @ A_27a )
= ( lam @ A_27a
@ ^ [V0x: $i] : ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__INN,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1n: tp__ty_2Enum_2Enum,V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V0e ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2b ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( ap @ V2b @ V0e ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2ESUB__BAG,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V2x ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V0b1 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V2x ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V1b2 ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EPSUB__BAG,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
& ( V0b1 != V1b2 ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__IN,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ V1b ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V0e ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V1b ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__UNION,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b ) @ V1c )
= ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V0b @ V2x ) ) @ ( ap @ V1c @ V2x ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__DIFF,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b1 ) @ V1b2 )
= ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ V0b1 @ V2x ) ) @ ( ap @ V1b2 @ V2x ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__INSERT,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ V1b )
= ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ V0e ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V1b @ V0e ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ V1b @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__cases,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( V0b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
| ? [V1b0: $i] :
( ( mem @ V1b0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ? [V2e: $i] :
( ( mem @ V2e @ A_27a )
& ( V0b
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e ) @ V1b0 ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__INTER,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__INTER @ A_27a ) @ V0b1 ) @ V1b2 )
= ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V0b1 @ V2x ) ) @ ( ap @ V1b2 @ V2x ) ) ) @ ( ap @ V0b1 @ V2x ) ) @ ( ap @ V1b2 @ V2x ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__MERGE,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__MERGE @ A_27a ) @ V0b1 ) @ V1b2 )
= ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V0b1 @ V2x ) ) @ ( ap @ V1b2 @ V2x ) ) ) @ ( ap @ V1b2 @ V2x ) ) @ ( ap @ V0b1 @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__MERGE__IDEM,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__MERGE @ A_27a ) @ V0b ) @ V0b )
= V0b ) ) ).
thf(conj_thm_2Ebag_2EBAG__INN__0,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V1e ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0b ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INN__LESS,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2n: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V1e ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V0b ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V1e ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ V0b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IN__BAG__INSERT,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e1: $i] :
( ( mem @ V1e1 @ A_27a )
=> ! [V2e2: $i] :
( ( mem @ V2e2 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1e1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e2 ) @ V0b ) ) )
<=> ( ( V1e1 = V2e2 )
| ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1e1 ) @ V0b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INN__BAG__INSERT,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2e1: $i] :
( ( mem @ V2e1 @ A_27a )
=> ! [V3e2: $i] :
( ( mem @ V3e2 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V2e1 ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V3e2 ) @ V1b ) ) )
<=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V2e1 ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V1b ) )
& ( V2e1 = V3e2 ) )
| ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V2e1 ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INN__BAG__INSERT__STRONG,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2e1: $i] :
( ( mem @ V2e1 @ A_27a )
=> ! [V3e2: $i] :
( ( mem @ V3e2 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V2e1 ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V3e2 ) @ V0b ) ) )
<=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V2e1 ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0b ) )
& ( V2e1 = V3e2 ) )
| ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V2e1 ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0b ) )
& ( V2e1 != V3e2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__UNION__EQ__LCANCEL1,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( V0b
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b ) @ V1c ) )
<=> ( V1c
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__UNION__EQ__RCANCEL1,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( V0b
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1c ) @ V0b ) )
<=> ( V1c
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IN__BAG__UNION,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e ) @ V0b1 ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e ) @ V1b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INN__BAG__UNION,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2b1: $i] :
( ( mem @ V2b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3b2: $i] :
( ( mem @ V3b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V1e ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V2b1 ) @ V3b2 ) ) )
<=> ? [V4m1: tp__ty_2Enum_2Enum,V5m2: tp__ty_2Enum_2Enum] :
( ( V0n
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V4m1 ) ) @ ( inj__ty_2Enum_2Enum @ V5m2 ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V1e ) @ ( inj__ty_2Enum_2Enum @ V4m1 ) ) @ V2b1 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V1e ) @ ( inj__ty_2Enum_2Enum @ V5m2 ) ) @ V3b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INN__BAG__MERGE,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2b1: $i] :
( ( mem @ V2b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3b2: $i] :
( ( mem @ V3b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V1e ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__MERGE @ A_27a ) @ V2b1 ) @ V3b2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V1e ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V2b1 ) )
| ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V1e ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V3b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IN__BAG__MERGE,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__MERGE @ A_27a ) @ V1b1 ) @ V2b2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ V1b1 ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ V2b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__EXTENSION,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( V0b1 = V1b2 )
<=> ! [V2n: tp__ty_2Enum_2Enum,V3e: $i] :
( ( mem @ V3e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V3e ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V0b1 ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V3e ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__UNION__INSERT,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ V1b1 ) ) @ V2b2 )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b1 ) @ V2b2 ) ) )
& ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ V2b2 ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b1 ) @ V2b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INSERT__DIFF,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ V1b )
!= V1b ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INSERT__NOT__EMPTY,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ V1b )
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INSERT__ONE__ONE,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2x ) @ V0b1 )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2x ) @ V1b2 ) )
<=> ( V0b1 = V1b2 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EC__BAG__INSERT__ONE__ONE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ V2b )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1y ) @ V2b ) )
<=> ( V0x = V1y ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INSERT__commutes,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e1: $i] :
( ( mem @ V1e1 @ A_27a )
=> ! [V2e2: $i] :
( ( mem @ V2e2 @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e2 ) @ V0b ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e2 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e1 ) @ V0b ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DECOMPOSE,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ V1b ) )
=> ? [V2b_27: $i] :
( ( mem @ V2b_27 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( V1b
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ V2b_27 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__UNION__LEFT__CANCEL,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b ) @ V1b1 )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b ) @ V2b2 ) )
<=> ( V1b1 = V2b2 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ECOMM__BAG__UNION,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b2 ) @ V0b1 ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__UNION__RIGHT__CANCEL,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b1 ) @ V0b )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V2b2 ) @ V0b ) )
<=> ( V1b1 = V2b2 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EASSOC__BAG__UNION,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b3: $i] :
( ( mem @ V2b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b2 ) @ V2b3 ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) ) @ V2b3 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__UNION__EMPTY,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ! [V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
= V0b ) )
& ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27b ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27b ) ) @ V1b )
= V1b ) )
& ! [V2b1: $i] :
( ( mem @ V2b1 @ ( arr @ A_27c @ ty_2Enum_2Enum ) )
=> ! [V3b2: $i] :
( ( mem @ V3b2 @ ( arr @ A_27c @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( 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(ax_thm_2Ebag_2EBAG__DELETE,axiom,
! [A_27a: del,V0b0: $i] :
( ( mem @ V0b0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b0 ) @ V1e ) @ V2b ) )
<=> ( V0b0
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DELETE__EMPTY,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) @ V0e ) @ V1b ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DELETE__commutes,axiom,
! [A_27a: del,V0b0: $i] :
( ( mem @ V0b0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3e1: $i] :
( ( mem @ V3e1 @ A_27a )
=> ! [V4e2: $i] :
( ( mem @ V4e2 @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b0 ) @ V3e1 ) @ V1b1 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V1b1 ) @ V4e2 ) @ V2b2 ) ) )
=> ? [V5b_27: $i] :
( ( mem @ V5b_27 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b0 ) @ V4e2 ) @ V5b_27 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V5b_27 ) @ V3e1 ) @ V2b2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DELETE__11,axiom,
! [A_27a: del,V0b0: $i] :
( ( mem @ V0b0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e1: $i] :
( ( mem @ V1e1 @ A_27a )
=> ! [V2e2: $i] :
( ( mem @ V2e2 @ A_27a )
=> ! [V3b1: $i] :
( ( mem @ V3b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V4b2: $i] :
( ( mem @ V4b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b0 ) @ V1e1 ) @ V3b1 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b0 ) @ V2e2 ) @ V4b2 ) ) )
=> ( ( V1e1 = V2e2 )
<=> ( V3b1 = V4b2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INN__BAG__DELETE,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V2e ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0b ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ? [V3b_27: $i] :
( ( mem @ V3b_27 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b ) @ V2e ) @ V3b_27 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IN__BAG__DELETE,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1e ) @ V0b ) )
=> ? [V2b_27: $i] :
( ( mem @ V2b_27 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b ) @ V1e ) @ V2b_27 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DELETE__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2b1: $i] :
( ( mem @ V2b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3b2: $i] :
( ( mem @ V3b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ V2b1 ) ) @ V1y ) @ V3b2 ) )
=> ( ( ( V0x = V1y )
& ( V2b1 = V3b2 ) )
| ( ? [V4b3: $i] :
( ( mem @ V4b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V2b1 ) @ V1y ) @ V4b3 ) ) )
& ( V0x != V1y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DELETE__BAG__IN__up,axiom,
! [A_27a: del,V0b0: $i] :
( ( mem @ V0b0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b0 ) @ V2e ) @ V1b ) )
=> ! [V3e_27: $i] :
( ( mem @ V3e_27 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V3e_27 ) @ V1b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V3e_27 ) @ V0b0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DELETE__BAG__IN__down,axiom,
! [A_27a: del,V0b0: $i] :
( ( mem @ V0b0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2e1: $i] :
( ( mem @ V2e1 @ A_27a )
=> ! [V3e2: $i] :
( ( mem @ V3e2 @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b0 ) @ V2e1 ) @ V1b ) )
& ( V2e1 != V3e2 )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V3e2 ) @ V0b0 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V3e2 ) @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DELETE__BAG__IN,axiom,
! [A_27a: del,V0b0: $i] :
( ( mem @ V0b0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b0 ) @ V2e ) @ V1b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e ) @ V0b0 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DELETE__concrete,axiom,
! [A_27a: del,V0b0: $i] :
( ( mem @ V0b0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b0 ) @ V2e ) @ V1b ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E @ ( ap @ V0b0 @ V2e ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
& ( V1b
= ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3x ) @ V2e ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ V0b0 @ V2e ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ V0b0 @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__UNION__DIFF__eliminate,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b ) @ V1c ) ) @ V1c )
= V0b )
& ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1c ) @ V0b ) ) @ V1c )
= V0b ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INSERT__EQUAL,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ! [V3N: $i] :
( ( mem @ V3N @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0a ) @ V1M )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2b ) @ V3N ) )
<=> ( ( ( V1M = V3N )
& ( V0a = V2b ) )
| ? [V4k: $i] :
( ( mem @ V4k @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( V1M
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2b ) @ V4k ) )
& ( V3N
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0a ) @ V4k ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DELETE__TWICE,axiom,
! [A_27a: del,V0b0: $i] :
( ( mem @ V0b0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e1: $i] :
( ( mem @ V1e1 @ A_27a )
=> ! [V2e2: $i] :
( ( mem @ V2e2 @ A_27a )
=> ! [V3b1: $i] :
( ( mem @ V3b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V4b2: $i] :
( ( mem @ V4b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b0 ) @ V1e1 ) @ V3b1 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b0 ) @ V2e2 ) @ V4b2 ) )
& ( V3b1 != V4b2 ) )
=> ? [V5b: $i] :
( ( mem @ V5b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V3b1 ) @ V2e2 ) @ V5b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V4b2 ) @ V1e1 ) @ V5b ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2ESING__BAG,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2ESING__BAG @ A_27a ) @ V0b ) )
<=> ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( V0b
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1x ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESING__BAG__THM,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ( p @ ( ap @ ( c_2Ebag_2ESING__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EEL__BAG,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ( ( ap @ ( c_2Ebag_2EEL__BAG @ A_27a ) @ V0e )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).
thf(conj_thm_2Ebag_2EEL__BAG__11,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( ( ap @ ( c_2Ebag_2EEL__BAG @ A_27a ) @ V0x )
= ( ap @ ( c_2Ebag_2EEL__BAG @ A_27a ) @ V1y ) )
=> ( V0x = V1y ) ) ) ) ).
thf(conj_thm_2Ebag_2EEL__BAG__INSERT__squeeze,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( ap @ ( c_2Ebag_2EEL__BAG @ A_27a ) @ V0x )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2y ) @ V1b ) )
=> ( ( V0x = V2y )
& ( V1b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESING__EL__BAG,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ( p @ ( ap @ ( c_2Ebag_2ESING__BAG @ A_27a ) @ ( ap @ ( c_2Ebag_2EEL__BAG @ A_27a ) @ V0e ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INSERT__UNION,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ V0b )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( c_2Ebag_2EEL__BAG @ A_27a ) @ V1e ) ) @ V0b ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INSERT__EQ__UNION,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ V3b )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b1 ) @ V2b2 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ V1b1 ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ V2b2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DELETE__SING,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b ) @ V1e ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
=> ( p @ ( ap @ ( c_2Ebag_2ESING__BAG @ A_27a ) @ V0b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ENOT__IN__EMPTY__BAG,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0x ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INN__EMPTY__BAG,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V0e ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
<=> ( V1n = fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Ebag_2EMEMBER__NOT__EMPTY,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1x ) @ V0b ) ) )
<=> ( V0b
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__LEQ,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0b1 @ V2x ) ) @ ( ap @ V1b2 @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__EMPTY,axiom,
! [A_27a: del] :
( ! [V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) @ V0b ) ) )
& ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1b ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
<=> ( V1b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__REFL,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b ) @ V0b ) ) ) ).
thf(conj_thm_2Ebag_2EPSUB__BAG__IRREFL,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ V0b ) @ V0b ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__ANTISYM,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1b2 ) @ V0b1 ) ) )
=> ( V0b1 = V1b2 ) ) ) ) ).
thf(conj_thm_2Ebag_2EPSUB__BAG__ANTISYM,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ~ ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ V1b2 ) @ V0b1 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__TRANS,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b3: $i] :
( ( mem @ V2b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1b2 ) @ V2b3 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V2b3 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EPSUB__BAG__TRANS,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b3: $i] :
( ( mem @ V2b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ V1b2 ) @ V2b3 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ V0b1 ) @ V2b3 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EPSUB__BAG__SUB__BAG,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EPSUB__BAG__NOT__EQ,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
=> ( V0b1 != V1b2 ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DIFF__EMPTY,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ! [V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b ) @ V0b )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
& ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27b ) @ V1b ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27b ) )
= V1b ) )
& ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27c @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27c ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27c ) ) @ V2b )
= ( c_2Ebag_2EEMPTY__BAG @ A_27c ) ) )
& ! [V3b1: $i] :
( ( mem @ V3b1 @ ( arr @ A_27d @ ty_2Enum_2Enum ) )
=> ! [V4b2: $i] :
( ( mem @ V4b2 @ ( arr @ A_27d @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27d ) @ V3b1 ) @ V4b2 ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27d ) @ V3b1 ) @ V4b2 )
= ( c_2Ebag_2EEMPTY__BAG @ A_27d ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DIFF__EMPTY__simple,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ! [V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b ) @ V0b )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
& ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27b ) @ V1b ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27b ) )
= V1b ) )
& ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27c @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27c ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27c ) ) @ V2b )
= ( c_2Ebag_2EEMPTY__BAG @ A_27c ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DIFF__EQ__EMPTY,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b ) @ V1c )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b ) @ V1c ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DIFF__INSERT__same,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ V1b1 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ V2b2 ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1b1 ) @ V2b2 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DIFF__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0x ) @ V1b1 ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ V2b2 ) ) @ V1b1 )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V2b2 ) @ V1b1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ENOT__IN__BAG__DIFF,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0x ) @ V1b1 ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ V2b2 ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1b1 ) @ V2b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IN__DIFF__I,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ V1b1 ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ V2b2 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1b1 ) @ V2b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IN__DIFF__E,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1b1 ) @ V2b2 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ V1b1 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__UNION__DIFF,axiom,
! [A_27a: del,V0X: $i] :
( ( mem @ V0X @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1Y: $i] :
( ( mem @ V1Y @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2Z: $i] :
( ( mem @ V2Z @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V2Z ) @ V1Y ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0X ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1Y ) @ V2Z ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0X ) @ V1Y ) ) @ V2Z ) )
& ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1Y ) @ V2Z ) ) @ V0X )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0X ) @ V1Y ) ) @ V2Z ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DIFF__2L,axiom,
! [A_27a: del,V0X: $i] :
( ( mem @ V0X @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1Y: $i] :
( ( mem @ V1Y @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2Z: $i] :
( ( mem @ V2Z @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0X ) @ V1Y ) ) @ V2Z )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0X ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1Y ) @ V2Z ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DIFF__2R,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2C: $i] :
( ( mem @ V2C @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V2C ) @ V1B ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0A ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1B ) @ V2C ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0A ) @ V2C ) ) @ V1B ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__BAG__DIFF,axiom,
! [A_27a: del,V0X: $i] :
( ( mem @ V0X @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1Y: $i] :
( ( mem @ V1Y @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2Y_27: $i] :
( ( mem @ V2Y_27 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3Z: $i] :
( ( mem @ V3Z @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V4W: $i] :
( ( mem @ V4W @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0X ) @ V1Y ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V3Z ) @ V4W ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0X ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1Y ) @ V2Y_27 ) ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V3Z ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V4W ) @ V2Y_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DIFF__UNION__eliminate,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b3: $i] :
( ( mem @ V2b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V2b3 ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1b2 ) @ V2b3 ) )
& ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V2b3 ) @ V0b1 ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1b2 ) @ V2b3 ) )
& ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b2 ) @ V0b1 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V2b3 ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1b2 ) @ V2b3 ) )
& ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b2 ) @ V0b1 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V2b3 ) @ V0b1 ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1b2 ) @ V2b3 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__UNION__eliminate,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b3: $i] :
( ( mem @ V2b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V2b3 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1b2 ) @ V2b3 ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V2b3 ) @ V0b1 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1b2 ) @ V2b3 ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b2 ) @ V0b1 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V2b3 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1b2 ) @ V2b3 ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b2 ) @ V0b1 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V2b3 ) @ V0b1 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1b2 ) @ V2b3 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emove__BAG__UNION__over__eq,axiom,
! [A_27a: del,V0X: $i] :
( ( mem @ V0X @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1Y: $i] :
( ( mem @ V1Y @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2Z: $i] :
( ( mem @ V2Z @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0X ) @ V1Y )
= V2Z )
=> ( V0X
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V2Z ) @ V1Y ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__UNION,axiom,
! [A_27a: del] :
( ! [V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b2 ) @ V2b ) ) ) ) ) ) )
& ! [V3b1: $i] :
( ( mem @ V3b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V4b2: $i] :
( ( mem @ V4b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V3b1 ) @ V4b2 ) )
=> ! [V5b: $i] :
( ( mem @ V5b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V3b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V5b ) @ V4b2 ) ) ) ) ) ) )
& ! [V6b1: $i] :
( ( mem @ V6b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V7b2: $i] :
( ( mem @ V7b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V8b3: $i] :
( ( mem @ V8b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V6b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V7b2 ) @ V8b3 ) ) )
=> ! [V9b: $i] :
( ( mem @ V9b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V6b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V7b2 ) @ V9b ) ) @ V8b3 ) ) ) ) ) ) ) )
& ! [V10b1: $i] :
( ( mem @ V10b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V11b2: $i] :
( ( mem @ V11b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V12b3: $i] :
( ( mem @ V12b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V10b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V11b2 ) @ V12b3 ) ) )
=> ! [V13b: $i] :
( ( mem @ V13b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V10b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V13b ) @ V11b2 ) ) @ V12b3 ) ) ) ) ) ) ) )
& ! [V14b1: $i] :
( ( mem @ V14b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V15b2: $i] :
( ( mem @ V15b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V16b3: $i] :
( ( mem @ V16b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V14b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V16b3 ) @ V15b2 ) ) )
=> ! [V17b: $i] :
( ( mem @ V17b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V14b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V16b3 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V15b2 ) @ V17b ) ) ) ) ) ) ) ) )
& ! [V18b1: $i] :
( ( mem @ V18b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V19b2: $i] :
( ( mem @ V19b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V20b3: $i] :
( ( mem @ V20b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V18b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V20b3 ) @ V19b2 ) ) )
=> ! [V21b: $i] :
( ( mem @ V21b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V18b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V20b3 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V21b ) @ V19b2 ) ) ) ) ) ) ) ) )
& ! [V22b1: $i] :
( ( mem @ V22b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V23b2: $i] :
( ( mem @ V23b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V24b3: $i] :
( ( mem @ V24b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V25b4: $i] :
( ( mem @ V25b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V22b1 ) @ V24b3 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V23b2 ) @ V25b4 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V22b1 ) @ V23b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V24b3 ) @ V25b4 ) ) ) ) ) ) ) ) )
& ! [V26b1: $i] :
( ( mem @ V26b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V27b2: $i] :
( ( mem @ V27b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V28b3: $i] :
( ( mem @ V28b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V29b4: $i] :
( ( mem @ V29b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V26b1 ) @ V29b4 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V27b2 ) @ V28b3 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V26b1 ) @ V27b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V28b3 ) @ V29b4 ) ) ) ) ) ) ) ) )
& ! [V30b1: $i] :
( ( mem @ V30b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V31b2: $i] :
( ( mem @ V31b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V32b3: $i] :
( ( mem @ V32b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V33b4: $i] :
( ( mem @ V33b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V34b5: $i] :
( ( mem @ V34b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V30b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V32b3 ) @ V34b5 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V31b2 ) @ V33b4 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V30b1 ) @ V31b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V32b3 ) @ V33b4 ) ) @ V34b5 ) ) ) ) ) ) ) ) ) )
& ! [V35b1: $i] :
( ( mem @ V35b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V36b2: $i] :
( ( mem @ V36b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V37b3: $i] :
( ( mem @ V37b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V38b4: $i] :
( ( mem @ V38b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V39b5: $i] :
( ( mem @ V39b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V35b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V38b4 ) @ V39b5 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V36b2 ) @ V37b3 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V35b1 ) @ V36b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V37b3 ) @ V38b4 ) ) @ V39b5 ) ) ) ) ) ) ) ) ) )
& ! [V40b1: $i] :
( ( mem @ V40b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V41b2: $i] :
( ( mem @ V41b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V42b3: $i] :
( ( mem @ V42b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V43b4: $i] :
( ( mem @ V43b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V44b5: $i] :
( ( mem @ V44b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V41b2 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V42b3 ) @ V44b5 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V40b1 ) @ V43b4 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V40b1 ) @ V41b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V42b3 ) @ V43b4 ) ) @ V44b5 ) ) ) ) ) ) ) ) ) )
& ! [V45b1: $i] :
( ( mem @ V45b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V46b2: $i] :
( ( mem @ V46b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V47b3: $i] :
( ( mem @ V47b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V48b4: $i] :
( ( mem @ V48b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V49b5: $i] :
( ( mem @ V49b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V46b2 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V48b4 ) @ V49b5 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V45b1 ) @ V47b3 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V45b1 ) @ V46b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V47b3 ) @ V48b4 ) ) @ V49b5 ) ) ) ) ) ) ) ) ) )
& ! [V50b1: $i] :
( ( mem @ V50b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V51b2: $i] :
( ( mem @ V51b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V52b3: $i] :
( ( mem @ V52b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V53b4: $i] :
( ( mem @ V53b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V54b5: $i] :
( ( mem @ V54b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V50b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V54b5 ) @ V52b3 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V51b2 ) @ V53b4 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V51b2 ) @ V50b1 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V54b5 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V52b3 ) @ V53b4 ) ) ) ) ) ) ) ) ) ) )
& ! [V55b1: $i] :
( ( mem @ V55b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V56b2: $i] :
( ( mem @ V56b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V57b3: $i] :
( ( mem @ V57b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V58b4: $i] :
( ( mem @ V58b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V59b5: $i] :
( ( mem @ V59b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V55b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V59b5 ) @ V58b4 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V56b2 ) @ V57b3 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V56b2 ) @ V55b1 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V59b5 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V57b3 ) @ V58b4 ) ) ) ) ) ) ) ) ) ) )
& ! [V60b1: $i] :
( ( mem @ V60b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V61b2: $i] :
( ( mem @ V61b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V62b3: $i] :
( ( mem @ V62b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V63b4: $i] :
( ( mem @ V63b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V64b5: $i] :
( ( mem @ V64b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V61b2 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V64b5 ) @ V62b3 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V60b1 ) @ V63b4 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V61b2 ) @ V60b1 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V64b5 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V62b3 ) @ V63b4 ) ) ) ) ) ) ) ) ) ) )
& ! [V65b1: $i] :
( ( mem @ V65b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V66b2: $i] :
( ( mem @ V66b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V67b3: $i] :
( ( mem @ V67b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V68b4: $i] :
( ( mem @ V68b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V69b5: $i] :
( ( mem @ V69b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V66b2 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V69b5 ) @ V68b4 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V65b1 ) @ V67b3 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V66b2 ) @ V65b1 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V69b5 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V67b3 ) @ V68b4 ) ) ) ) ) ) ) ) ) ) )
& ! [V70b1: $i] :
( ( mem @ V70b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V71b2: $i] :
( ( mem @ V71b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V72b3: $i] :
( ( mem @ V72b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V73b4: $i] :
( ( mem @ V73b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V74b5: $i] :
( ( mem @ V74b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V70b1 ) @ V71b2 ) ) @ V73b4 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V72b3 ) @ V74b5 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V70b1 ) @ V72b3 ) ) @ V71b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V73b4 ) @ V74b5 ) ) ) ) ) ) ) ) ) )
& ! [V75b1: $i] :
( ( mem @ V75b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V76b2: $i] :
( ( mem @ V76b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V77b3: $i] :
( ( mem @ V77b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V78b4: $i] :
( ( mem @ V78b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V79b5: $i] :
( ( mem @ V79b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V75b1 ) @ V76b2 ) ) @ V79b5 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V77b3 ) @ V78b4 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V75b1 ) @ V77b3 ) ) @ V76b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V78b4 ) @ V79b5 ) ) ) ) ) ) ) ) ) )
& ! [V80b1: $i] :
( ( mem @ V80b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V81b2: $i] :
( ( mem @ V81b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V82b3: $i] :
( ( mem @ V82b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V83b4: $i] :
( ( mem @ V83b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V84b5: $i] :
( ( mem @ V84b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V82b3 ) @ V81b2 ) ) @ V83b4 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V80b1 ) @ V84b5 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V80b1 ) @ V82b3 ) ) @ V81b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V83b4 ) @ V84b5 ) ) ) ) ) ) ) ) ) )
& ! [V85b1: $i] :
( ( mem @ V85b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V86b2: $i] :
( ( mem @ V86b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V87b3: $i] :
( ( mem @ V87b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V88b4: $i] :
( ( mem @ V88b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V89b5: $i] :
( ( mem @ V89b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V87b3 ) @ V86b2 ) ) @ V89b5 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V85b1 ) @ V88b4 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V85b1 ) @ V87b3 ) ) @ V86b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V88b4 ) @ V89b5 ) ) ) ) ) ) ) ) ) )
& ! [V90b1: $i] :
( ( mem @ V90b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V91b2: $i] :
( ( mem @ V91b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V92b3: $i] :
( ( mem @ V92b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V93b4: $i] :
( ( mem @ V93b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V94b5: $i] :
( ( mem @ V94b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V91b2 ) @ V90b1 ) ) @ V93b4 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V92b3 ) @ V94b5 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V91b2 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V90b1 ) @ V92b3 ) ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V94b5 ) @ V93b4 ) ) ) ) ) ) ) ) ) )
& ! [V95b1: $i] :
( ( mem @ V95b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V96b2: $i] :
( ( mem @ V96b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V97b3: $i] :
( ( mem @ V97b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V98b4: $i] :
( ( mem @ V98b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V99b5: $i] :
( ( mem @ V99b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V96b2 ) @ V95b1 ) ) @ V99b5 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V97b3 ) @ V98b4 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V96b2 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V95b1 ) @ V97b3 ) ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V99b5 ) @ V98b4 ) ) ) ) ) ) ) ) ) )
& ! [V100b1: $i] :
( ( mem @ V100b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V101b2: $i] :
( ( mem @ V101b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V102b3: $i] :
( ( mem @ V102b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V103b4: $i] :
( ( mem @ V103b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V104b5: $i] :
( ( mem @ V104b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V101b2 ) @ V102b3 ) ) @ V103b4 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V100b1 ) @ V104b5 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V101b2 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V100b1 ) @ V102b3 ) ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V104b5 ) @ V103b4 ) ) ) ) ) ) ) ) ) )
& ! [V105b1: $i] :
( ( mem @ V105b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V106b2: $i] :
( ( mem @ V106b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V107b3: $i] :
( ( mem @ V107b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V108b4: $i] :
( ( mem @ V108b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V109b5: $i] :
( ( mem @ V109b5 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V106b2 ) @ V107b3 ) ) @ V109b5 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V105b1 ) @ V108b4 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V106b2 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V105b1 ) @ V107b3 ) ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V109b5 ) @ V108b4 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__EL__BAG,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( c_2Ebag_2EEL__BAG @ A_27a ) @ V0e ) ) @ V1b ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__INSERT,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ V1b1 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ V2b2 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1b1 ) @ V2b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__INSERT__I,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b ) @ V1c ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e ) @ V1c ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ENOT__IN__SUB__BAG__INSERT,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e ) @ V0b1 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e ) @ V1b2 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__BAG__IN,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ V1b1 ) ) @ V2b2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0x ) @ V2b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__EXISTS,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
<=> ? [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( V1b2
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__UNION__DIFF,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b3: $i] :
( ( mem @ V2b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V2b3 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1b2 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V2b3 ) @ V0b1 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) ) @ V2b3 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__UNION__down,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b3: $i] :
( ( mem @ V2b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) ) @ V2b3 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V2b3 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1b2 ) @ V2b3 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__DIFF,axiom,
! [A_27a: del] :
( ! [V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
=> ! [V2b3: $i] :
( ( mem @ V2b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b1 ) @ V2b3 ) ) @ V1b2 ) ) ) ) ) )
& ! [V3b1: $i] :
( ( mem @ V3b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V4b2: $i] :
( ( mem @ V4b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V5b3: $i] :
( ( mem @ V5b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V6b4: $i] :
( ( mem @ V6b4 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V4b2 ) @ V3b1 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V6b4 ) @ V5b3 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V3b1 ) @ V4b2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V5b3 ) @ V6b4 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V3b1 ) @ V6b4 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V4b2 ) @ V5b3 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__PSUB__BAG,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
| ( V0b1 = V1b2 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DELETE__PSUB__BAG,axiom,
! [A_27a: del,V0b0: $i] :
( ( mem @ V0b0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DELETE @ A_27a ) @ V0b0 ) @ V1e ) @ V2b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ V2b ) @ V0b0 ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2ESET__OF__BAG,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V0b )
= ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1x ) @ V0b ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__OF__SET,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Ebag_2EBAG__OF__SET @ A_27a ) @ V0P )
= ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0P ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESET__OF__EMPTY,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Ebag_2EBAG__OF__SET @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ).
thf(conj_thm_2Ebag_2EBAG__IN__BAG__OF__SET,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1p ) @ ( ap @ ( c_2Ebag_2EBAG__OF__SET @ A_27a ) @ V0P ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1p ) @ V0P ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__OF__EMPTY,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(conj_thm_2Ebag_2ESET__BAG__I,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ ( ap @ ( c_2Ebag_2EBAG__OF__SET @ A_27a ) @ V0s ) )
= V0s ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__SET,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V0b1 ) ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V1b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESET__OF__BAG__UNION,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V0b1 ) ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V1b2 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESET__OF__BAG__MERGE,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__MERGE @ A_27a ) @ V0b1 ) @ V1b2 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V0b1 ) ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V1b2 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESET__OF__BAG__INSERT,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ V1b ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0e ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESET__OF__EL__BAG,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ( ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ ( ap @ ( c_2Ebag_2EEL__BAG @ A_27a ) @ V0e ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0e ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Ebag_2ESET__OF__BAG__DIFF__SUBSET__down,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V0b1 ) ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V1b2 ) ) ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b1 ) @ V1b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESET__OF__BAG__DIFF__SUBSET__up,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b1 ) @ V1b2 ) ) ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V0b1 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EIN__SET__OF__BAG,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V1b ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0x ) @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESET__OF__BAG__EQ__EMPTY,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ( c_2Epred__set_2EEMPTY @ A_27a )
= ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V0b ) )
<=> ( V0b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
& ( ( ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V0b )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( V0b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__DISJOINT,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ V0b1 ) @ V1b2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V0b1 ) ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V1b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DISJOINT__EMPTY,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ V0b ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) @ V0b ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DISJOINT__DIFF,axiom,
! [A_27a: del,V0B1: $i] :
( ( mem @ V0B1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1B2: $i] :
( ( mem @ V1B2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0B1 ) @ V1B2 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1B2 ) @ V0B1 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DISJOINT__BAG__IN,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ V0b1 ) @ V1b2 ) )
<=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e ) @ V0b1 ) )
| ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e ) @ V1b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DISJOINT__BAG__INSERT,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2e1: $i] :
( ( mem @ V2e1 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e1 ) @ V0b1 ) ) @ V1b2 ) )
<=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e1 ) @ V1b2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ V0b1 ) @ V1b2 ) ) ) ) ) ) )
& ! [V3b1: $i] :
( ( mem @ V3b1 @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ! [V4b2: $i] :
( ( mem @ V4b2 @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ! [V5e2: $i] :
( ( mem @ V5e2 @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27b ) @ V3b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27b ) @ V5e2 ) @ V4b2 ) ) )
<=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27b ) @ V5e2 ) @ V3b1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27b ) @ V3b1 ) @ V4b2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DISJOINT__BAG__UNION,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b3: $i] :
( ( mem @ V2b3 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ V0b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b2 ) @ V2b3 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ V0b1 ) @ V1b2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ V0b1 ) @ V2b3 ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) ) @ V2b3 ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ V0b1 ) @ V2b3 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ V1b2 ) @ V2b3 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EFINITE__BAG,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
<=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ( ( ( p @ ( ap @ V1P @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
& ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ V1P @ V2b ) )
=> ! [V3e: $i] :
( ( mem @ V3e @ A_27a )
=> ( p @ ( ap @ V1P @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V3e ) @ V2b ) ) ) ) ) ) )
=> ( p @ ( ap @ V1P @ V0b ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EFINITE__EMPTY__BAG,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ).
thf(conj_thm_2Ebag_2EFINITE__BAG__INSERT,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ V0b ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EFINITE__BAG__INDUCT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
& ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ V0P @ V1b ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e ) @ V1b ) ) ) ) ) ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V3b ) )
=> ( p @ ( ap @ V0P @ V3b ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESTRONG__FINITE__BAG__INDUCT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
& ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b ) )
& ( p @ ( ap @ V0P @ V1b ) ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e ) @ V1b ) ) ) ) ) ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V3b ) )
=> ( p @ ( ap @ V0P @ V3b ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EFINITE__BAG__THM,axiom,
! [A_27a: del] :
( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
& ! [V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ V1b ) ) )
<=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EFINITE__BAG__DIFF,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b1 ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b1 ) @ V1b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EFINITE__BAG__DIFF__dual,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b1 ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1b2 ) @ V0b1 ) ) )
=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EFINITE__BAG__UNION,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) ) )
<=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EFINITE__SUB__BAG,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b1 ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1b2 ) @ V0b1 ) )
=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b2 ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__CARD__RELn,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__CARD__RELn @ A_27a ) @ V0b ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ V2P @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
& ! [V3b: $i] :
( ( mem @ V3b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ V2P @ V3b ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ! [V5e: $i] :
( ( mem @ V5e @ A_27a )
=> ( p @ ( ap @ ( ap @ V2P @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V5e ) @ V3b ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V2P @ V0b ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__CARD,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__CARD__RELn @ A_27a ) @ V0b ) @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V0b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__CARD__EMPTY,axiom,
! [A_27a: del] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
= fo__c_2Enum_2E0 ) ).
thf(conj_thm_2Ebag_2EBCARD__0,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V0b ) )
= fo__c_2Enum_2E0 )
<=> ( V0b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__CARD__THM,axiom,
! [A_27a: del] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
= fo__c_2Enum_2E0 )
& ! [V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ V0b ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V0b ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__CARD__UNION,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b2 ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V0b1 ) ) @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V1b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBCARD__SUC,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V0b ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ? [V2b0: $i] :
( ( mem @ V2b0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ? [V3e: $i] :
( ( mem @ V3e @ A_27a )
& ( V0b
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V3e ) @ V2b0 ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V2b0 ) )
= V1n ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__CARD__BAG__INN,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V2e ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0b ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V0b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__DIFF__EQ,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
=> ( V1b2
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1b2 ) @ V0b1 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__CARD,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V0b1 ) ) @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V1b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__CARD__DIFF,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1c ) @ V0b ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b ) @ V1c ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V0b ) ) @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V1c ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__FILTER__DEF,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V0P ) @ V1b )
= ( lam @ A_27a
@ ^ [V2e: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ V0P @ V2e ) ) @ ( ap @ V1b @ V2e ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__FILTER__EMPTY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V0P ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__FILTER__BAG__INSERT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ V2b ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ V0P @ V1e ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V0P ) @ V2b ) ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V0P ) @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EFINITE__BAG__FILTER,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b ) )
=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V0P ) @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INN__BAG__FILTER,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1n: tp__ty_2Enum_2Enum,V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V0e ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V2P ) @ V3b ) ) )
<=> ( ( V1n = fo__c_2Enum_2E0 )
| ( ( p @ ( ap @ V2P @ V0e ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V0e ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V3b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IN__BAG__FILTER,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V1P ) @ V2b ) ) )
<=> ( ( p @ ( ap @ V1P @ V0e ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__FILTER__FILTER,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V1Q ) @ V2b ) )
= ( ap
@ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V3a: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V3a ) ) @ ( ap @ V1Q @ V3a ) ) ) )
@ V2b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__FILTER__SUB__BAG,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V0P ) @ V1b ) ) @ V1b ) ) ) ) ).
thf(conj_thm_2Ebag_2ESET__OF__BAG__EQ__INSERT,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1e ) @ V2s )
= ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V0b ) )
<=> ? [V3b0: $i] :
( ( mem @ V3b0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ? [V4eb: $i] :
( ( mem @ V4eb @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( V0b
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V4eb ) @ V3b0 ) )
& ( V2s
= ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V3b0 ) )
& ! [V5e_27: $i] :
( ( mem @ V5e_27 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V5e_27 ) @ V4eb ) )
=> ( V5e_27 = V1e ) ) )
& ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1e ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1e ) @ V4eb ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EFINITE__SET__OF__BAG,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V0b ) ) )
<=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__IMAGE__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27b ) @ V0f ) @ V1b )
= ( lam @ A_27a
@ ^ [V2e: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( arr @ A_27b @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum )
@ ( lam @ ( arr @ A_27b @ ty_2Enum_2Enum )
@ ^ [V3sb: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27b ) @ V3sb ) ) @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27b ) @ V3sb ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( ap
@ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27b )
@ ( lam @ A_27b
@ ^ [V4e0: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V0f @ V4e0 ) ) @ V2e ) ) )
@ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IMAGE__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a ) @ V0f ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
= ( c_2Ebag_2EEMPTY__BAG @ A_27b ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IMAGE__FINITE__INSERT,axiom,
! [A_27a: del,A_27b: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a ) @ V1f ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e ) @ V0b ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27b ) @ ( ap @ V1f @ V2e ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a ) @ V1f ) @ V0b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IMAGE__FINITE__UNION,axiom,
! [A_27a: del,A_27b: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b2 ) ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a ) @ V2f ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27b ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a ) @ V2f ) @ V0b1 ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a ) @ V2f ) @ V1b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IMAGE__FINITE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b ) )
=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27b ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a ) @ V0f ) @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IMAGE__COMPOSE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27c @ A_27a ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27c @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27c ) @ V2b ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a ) @ V0f ) @ V1g ) ) @ V2b )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27c ) @ V1g ) @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IMAGE__FINITE__I,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27a ) @ ( c_2Ecombin_2EI @ A_27a ) ) @ V0b )
= V0b ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IN__FINITE__BAG__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27b )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27b ) @ V1x ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a ) @ V2f ) @ V0b ) ) )
<=> ? [V3y: $i] :
( ( mem @ V3y @ A_27a )
& ( ( ap @ V2f @ V3y )
= V1x )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V3y ) @ V0b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IMAGE__EQ__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27b @ A_27a ) @ V1f ) @ V0b )
= ( c_2Ebag_2EEMPTY__BAG @ A_27b ) )
<=> ( V0b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__CHOICE__DEF,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( V0b
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ ( ap @ ( c_2Ebag_2EBAG__CHOICE @ A_27a ) @ V0b ) ) @ V0b ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__REST__DEF,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ V0b )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b ) @ ( ap @ ( c_2Ebag_2EEL__BAG @ A_27a ) @ ( ap @ ( c_2Ebag_2EBAG__CHOICE @ A_27a ) @ V0b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INSERT__CHOICE__REST,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( V0b
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
=> ( V0b
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ ( ap @ ( c_2Ebag_2EBAG__CHOICE @ A_27a ) @ V0b ) ) @ ( ap @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ V0b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__CHOICE__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Ebag_2EBAG__CHOICE @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
= V0x ) ) ).
thf(conj_thm_2Ebag_2EBAG__REST__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__REST,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ V0b ) ) @ V0b ) ) ) ).
thf(conj_thm_2Ebag_2EPSUB__BAG__REST,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( V0b
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ ( ap @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ V0b ) ) @ V0b ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__UNION__MONO,axiom,
! [A_27a: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0x ) @ V1y ) ) ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V3y ) @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EPSUB__BAG__CARD,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EPSUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V0b1 ) ) @ ( ap @ ( c_2Ebag_2EBAG__CARD @ A_27a ) @ V1b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EEL__BAG__BAG__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1y ) @ V2b ) )
<=> ( ( V0x = V1y )
& ( V2b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EEL__BAG__SUB__BAG,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0x ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) @ V1b ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0x ) @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EITBAG__IND,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27b @ bool ) ) )
=> ( ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3acc: $i] :
( ( mem @ V3acc @ A_27b )
=> ( ( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V2b ) )
& ( V2b
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ ( ap @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ V2b ) ) @ ( ap @ ( ap @ V0f @ ( ap @ ( c_2Ebag_2EBAG__CHOICE @ A_27a ) @ V2b ) ) @ V3acc ) ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V2b ) @ V3acc ) ) ) ) )
=> ! [V4v: $i] :
( ( mem @ V4v @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V5v1: $i] :
( ( mem @ V5v1 @ A_27b )
=> ( p @ ( ap @ ( ap @ V1P @ V4v ) @ V5v1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EITBAG__THM,axiom,
! [A_27a: del,A_27b: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V2acc: $i] :
( ( mem @ V2acc @ A_27b )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EITBAG @ A_27a @ A_27b ) @ V1f ) @ V0b ) @ V2acc )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0b ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) @ V2acc ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EITBAG @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ V0b ) ) @ ( ap @ ( ap @ V1f @ ( ap @ ( c_2Ebag_2EBAG__CHOICE @ A_27a ) @ V0b ) ) @ V2acc ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EITBAG__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1acc: $i] :
( ( mem @ V1acc @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EITBAG @ A_27a @ A_27b ) @ V0f ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) @ V1acc )
= V1acc ) ) ) ).
thf(conj_thm_2Ebag_2EITBAG__INSERT,axiom,
! [A_27a: del,A_27b: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V3acc: $i] :
( ( mem @ V3acc @ A_27b )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EITBAG @ A_27a @ A_27b ) @ V2f ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1x ) @ V0b ) ) @ V3acc )
= ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EITBAG @ A_27a @ A_27b ) @ V2f ) @ ( ap @ ( c_2Ebag_2EBAG__REST @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1x ) @ V0b ) ) ) @ ( ap @ ( ap @ V2f @ ( ap @ ( c_2Ebag_2EBAG__CHOICE @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1x ) @ V0b ) ) ) @ V3acc ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ECOMMUTING__ITBAG__INSERT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4z: $i] :
( ( mem @ V4z @ A_27b )
=> ( ( ap @ ( ap @ V0f @ V2x ) @ ( ap @ ( ap @ V0f @ V3y ) @ V4z ) )
= ( ap @ ( ap @ V0f @ V3y ) @ ( ap @ ( ap @ V0f @ V2x ) @ V4z ) ) ) ) ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6a: $i] :
( ( mem @ V6a @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EITBAG @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V5x ) @ V1b ) ) @ V6a )
= ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EITBAG @ A_27a @ A_27b ) @ V0f ) @ V1b ) @ ( ap @ ( ap @ V0f @ V5x ) @ V6a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ECOMMUTING__ITBAG__RECURSES,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ! [V6z: $i] :
( ( mem @ V6z @ A_27b )
=> ( ( ap @ ( ap @ V0f @ V4x ) @ ( ap @ ( ap @ V0f @ V5y ) @ V6z ) )
= ( ap @ ( ap @ V0f @ V5y ) @ ( ap @ ( ap @ V0f @ V4x ) @ V6z ) ) ) ) ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V2b ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EITBAG @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ V2b ) ) @ V3a )
= ( ap @ ( ap @ V0f @ V1e ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EITBAG @ A_27a @ A_27b ) @ V0f ) @ V2b ) @ V3a ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__GEN__SUM__def,axiom,
! [V0bag: $i] :
( ( mem @ V0bag @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__SUM @ V0bag ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EITBAG @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_2B ) @ V0bag ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__GEN__PROD__def,axiom,
! [V0bag: $i] :
( ( mem @ V0bag @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ V0bag ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EITBAG @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_2A ) @ V0bag ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__GEN__SUM__EMPTY,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__SUM @ ( c_2Ebag_2EEMPTY__BAG @ ty_2Enum_2Enum ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= V0n ) ).
thf(conj_thm_2Ebag_2EBAG__GEN__PROD__EMPTY,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ ( c_2Ebag_2EEMPTY__BAG @ ty_2Enum_2Enum ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= V0n ) ).
thf(conj_thm_2Ebag_2EBAG__GEN__SUM__TAILREC,axiom,
! [V0b: $i] :
( ( mem @ V0b @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V0b ) )
=> ! [V1x: tp__ty_2Enum_2Enum,V2a: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__SUM @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__SUM @ V0b ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__GEN__SUM__REC,axiom,
! [V0b: $i] :
( ( mem @ V0b @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V0b ) )
=> ! [V1x: tp__ty_2Enum_2Enum,V2a: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__SUM @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__SUM @ V0b ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__GEN__PROD__TAILREC,axiom,
! [V0b: $i] :
( ( mem @ V0b @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V0b ) )
=> ! [V1x: tp__ty_2Enum_2Enum,V2a: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ V0b ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__GEN__PROD__REC,axiom,
! [V0b: $i] :
( ( mem @ V0b @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V0b ) )
=> ! [V1x: tp__ty_2Enum_2Enum,V2a: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ V0b ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__GEN__PROD__EQ__1,axiom,
! [V0b: $i] :
( ( mem @ V0b @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V0b ) )
=> ! [V1e: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ V0b ) @ ( inj__ty_2Enum_2Enum @ V1e ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
=> ( V1e
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__GEN__PROD__ALL__ONES,axiom,
! [V0b: $i] :
( ( mem @ V0b @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V0b ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ V0b ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
=> ! [V1x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V0b ) )
=> ( V1x
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__GEN__PROD__POSITIVE,axiom,
! [V0b: $i] :
( ( mem @ V0b @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V0b ) )
=> ( ! [V1x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V0b ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ V0b ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__EVERY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27a ) @ V0P ) @ V1b ) )
<=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e ) @ V1b ) )
=> ( p @ ( ap @ V0P @ V2e ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__EVERY__THM,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27a ) @ V0P ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) )
& ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27b )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27b ) @ V1P ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27b ) @ V2e ) @ V3b ) ) )
<=> ( ( p @ ( ap @ V1P @ V2e ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27b ) @ V1P ) @ V3b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__EVERY__UNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b1 ) @ V2b2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27a ) @ V0P ) @ V1b1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27a ) @ V0P ) @ V2b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__EVERY__MERGE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__MERGE @ A_27a ) @ V1b1 ) @ V2b2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27a ) @ V0P ) @ V1b1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27a ) @ V0P ) @ V2b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__EVERY__SET,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27a ) @ V0P ) @ V1b ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V1b ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ V0P @ V2x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__FILTER__EQ__EMPTY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__FILTER @ A_27a ) @ V0P ) @ V1b )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V0P ) ) @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESET__OF__BAG__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ( ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27b ) @ V0f ) @ V1b ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27b ) @ V1b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IMAGE__FINITE__RESTRICTED__I,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebag_2EBAG__EVERY @ A_27a )
@ ( lam @ A_27a
@ ^ [V2e: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V0f @ V2e ) ) @ V2e ) ) )
@ V1b ) ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__IMAGE @ A_27a @ A_27a ) @ V0f ) @ V1b )
= V1b ) ) ) ) ).
thf(ax_thm_2Ebag_2EBAG__ALL__DISTINCT,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V0b ) )
<=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0b @ V1e ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__ALL__DISTINCT__THM,axiom,
! [A_27a: del,A_27b: del] :
( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
& ! [V0e: $i] :
( ( mem @ V0e @ A_27b )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27b ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27b ) @ V0e ) @ V1b ) ) )
<=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27b ) @ V0e ) @ V1b ) )
& ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27b ) @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__ALL__DISTINCT__BAG__MERGE,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__MERGE @ A_27a ) @ V0b1 ) @ V1b2 ) ) )
<=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V0b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V1b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__ALL__DISTINCT__BAG__UNION,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b1 ) @ V1b2 ) ) )
<=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V0b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V1b2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ V0b1 ) @ V1b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__ALL__DISTINCT__DIFF,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V0b1 ) )
=> ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b1 ) @ V1b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__ALL__DISTINCT__DELETE,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V0b ) )
<=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1e ) @ V0b ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1e ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__ALL__DISTINCT__SET,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V0b ) )
<=> ( ( ap @ ( c_2Ebag_2EBAG__OF__SET @ A_27a ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V0b ) )
= V0b ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__ALL__DISTINCT__BAG__OF__SET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ ( ap @ ( c_2Ebag_2EBAG__OF__SET @ A_27a ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IN__BAG__DIFF__ALL__DISTINCT,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V0b1 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b1 ) @ V1b2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e ) @ V0b1 ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e ) @ V1b2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__ALL__DISTINCT,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V0b1 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b1 ) @ V1b2 ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2x ) @ V0b1 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2x ) @ V1b2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__ALL__DISTINCT__BAG__INN,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V0b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INN @ A_27a ) @ V2e ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0b ) )
<=> ( ( V1n = fo__c_2Enum_2E0 )
| ( ( V1n
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2e ) @ V0b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__ALL__DISTINCT__EXTENSION,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V0b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EBAG__ALL__DISTINCT @ A_27a ) @ V1b2 ) ) )
=> ( ( V0b1 = V1b2 )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2x ) @ V0b1 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V2x ) @ V1b2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ENOT__BAG__IN,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ V0b @ V1x ) )
= fo__c_2Enum_2E0 )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1x ) @ V0b ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__UNION__EQ__LEFT,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2d: $i] :
( ( mem @ V2d @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b ) @ V1c )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V0b ) @ V2d ) )
=> ( V1c = V2d ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IN__DIVIDES,axiom,
! [V0b: $i] :
( ( mem @ V0b @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1x: tp__ty_2Enum_2Enum,V2a: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V0b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V0b ) ) )
=> ( p @ ( ap @ ( ap @ c_2Edivides_2Edivides @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ V0b ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__GEN__PROD__UNION__LEM,axiom,
! [V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V0b1 ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V2a: tp__ty_2Enum_2Enum,V3b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V1b2 ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ ty_2Enum_2Enum ) @ V0b1 ) @ V1b2 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V2a ) ) @ ( inj__ty_2Enum_2Enum @ V3b ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ V0b1 ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) ) @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ V1b2 ) @ ( inj__ty_2Enum_2Enum @ V3b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__GEN__PROD__UNION,axiom,
! [V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V0b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V1b2 ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ ty_2Enum_2Enum ) @ V0b1 ) @ V1b2 ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ V0b1 ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ebag_2EBAG__GEN__PROD @ V1b2 ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2EBIG__BAG__UNION__def,axiom,
! [A_27a: del,V0sob: $i] :
( ( mem @ V0sob @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ( ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ V0sob )
= ( lam @ A_27a
@ ^ [V1x: $i] :
( ap
@ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
@ ( lam @ ( arr @ A_27a @ ty_2Enum_2Enum )
@ ^ [V2b: $i] : ( ap @ V2b @ V1x ) ) )
@ V0sob ) ) ) ) ).
thf(conj_thm_2Ebag_2EBIG__BAG__UNION__EMPTY,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ).
thf(conj_thm_2Ebag_2EBIG__BAG__UNION__INSERT,axiom,
! [A_27a: del,V0sob: $i] :
( ( mem @ V0sob @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0sob ) )
=> ( ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V1b ) @ V0sob ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b ) @ ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0sob ) @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBIG__BAG__UNION__DELETE,axiom,
! [A_27a: del,V0sob: $i] :
( ( mem @ V0sob @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0sob ) )
=> ( ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0sob ) @ V1b ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V1b ) @ V0sob ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ V0sob ) ) @ V1b ) ) @ ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ V0sob ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBIG__BAG__UNION__ITSET__BAG__UNION,axiom,
! [A_27a: del,V0sob: $i] :
( ( mem @ V0sob @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0sob ) )
=> ( ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ V0sob )
= ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( c_2Ebag_2EBAG__UNION @ A_27a ) ) @ V0sob ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EFINITE__BIG__BAG__UNION,axiom,
! [A_27a: del,V0sob: $i] :
( ( mem @ V0sob @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0sob ) )
& ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V1b ) @ V0sob ) )
=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b ) ) ) ) )
=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ V0sob ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__IN__BIG__BAG__UNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0P ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1e ) @ ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ V0P ) ) )
<=> ? [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V1e ) @ V2b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V2b ) @ V0P ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBIG__BAG__UNION__EQ__EMPTY__BAG,axiom,
! [A_27a: del,V0sob: $i] :
( ( mem @ V0sob @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0sob ) )
=> ( ( ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ V0sob )
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
<=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V1b ) @ V0sob ) )
=> ( V1b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBIG__BAG__UNION__UNION,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0s1 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V1s2 ) ) )
=> ( ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0s1 ) @ V1s2 ) )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ V0s1 ) ) @ ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ V1s2 ) ) ) @ ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0s1 ) @ V1s2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBIG__BAG__UNION__EQ__ELEMENT,axiom,
! [A_27a: del,V0sob: $i] :
( ( mem @ V0sob @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V0sob ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V1b ) @ V0sob ) ) )
=> ( ( ( ap @ ( c_2Ebag_2EBIG__BAG__UNION @ A_27a ) @ V0sob )
= V1b )
<=> ! [V2b_27: $i] :
( ( mem @ V2b_27 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ V2b_27 ) @ V0sob ) )
=> ( ( V2b_27 = V1b )
| ( V2b_27
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2Emlt1__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0r ) @ V1b1 ) @ V2b2 ) )
<=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V2b2 ) )
& ? [V3e: $i] :
( ( mem @ V3e @ A_27a )
& ? [V4rep: $i] :
( ( mem @ V4rep @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ? [V5res: $i] :
( ( mem @ V5res @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( V1b1
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V4rep ) @ V5res ) )
& ( V2b2
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V5res ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V3e ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) )
& ! [V6e_27: $i] :
( ( mem @ V6e_27 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V6e_27 ) @ V4rep ) )
=> ( p @ ( ap @ ( ap @ V0r @ V6e_27 ) @ V3e ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__NOT__LESS__EMPTY,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0r ) @ V1b ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ENOT__mlt__EMPTY,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V1b ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__LESS__ADD,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1N: $i] :
( ( mem @ V1N @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2M0: $i] :
( ( mem @ V2M0 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0r ) @ V1N ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V2M0 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V3a ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) )
=> ( ? [V4M: $i] :
( ( mem @ V4M @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0r ) @ V4M ) @ V2M0 ) )
& ( V1N
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V4M ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V3a ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) )
| ? [V5KK: $i] :
( ( mem @ V5KK @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ! [V6b: $i] :
( ( mem @ V6b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V6b ) @ V5KK ) )
=> ( p @ ( ap @ ( ap @ V0r @ V6b ) @ V3a ) ) ) )
& ( V1N
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V2M0 ) @ V5KK ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt1__all__accessible,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Erelation_2EWFP @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V1M ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EWF__mlt1,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ETC__mlt1__FINITE__BAG,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V1b1 ) @ V2b2 ) )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V2b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ETC__mlt1__UNION2__I,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b1: $i] :
( ( mem @ V2b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b2 ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V2b1 ) )
& ( V1b2
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V2b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V2b1 ) @ V1b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ETC__mlt1__UNION1__I,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b1: $i] :
( ( mem @ V2b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b2 ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V2b1 ) )
& ( V2b1
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V1b2 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V2b1 ) @ V1b2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt__NOT__REFL,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V1a ) @ V1a ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt__INSERT__CANCEL__I,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V2a ) @ V3b ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ V2a ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ V3b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt1__INSERT__CANCEL,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ V2a ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ V3b ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) @ V2a ) @ V3b ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ebag_2Edominates__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
=> ! [V2s2: $i] :
( ( mem @ V2s2 @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Edominates @ A_27a @ A_27b ) @ V0R ) @ V1s1 ) @ V2s2 ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s1 ) )
=> ? [V4y: $i] :
( ( mem @ V4y @ A_27b )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4y ) @ V2s2 ) )
& ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Edominates__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Edominates @ A_27a @ A_27b ) @ V0R ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1b ) ) ) ) ).
thf(conj_thm_2Ebag_2Edominates__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Y: $i] :
( ( mem @ V1Y @ ( arr @ A_27a @ bool ) )
=> ! [V2X: $i] :
( ( mem @ V2X @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1Y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Edominates @ A_27a @ A_27a ) @ V0R ) @ V1Y ) @ V2X ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2X ) @ V1Y ) )
& ( V2X
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2X ) )
& ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt__dominates__thm1,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V1b1 ) @ V2b2 ) )
<=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V2b2 ) )
& ? [V3x: $i] :
( ( mem @ V3x @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ? [V4y: $i] :
( ( mem @ V4y @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( V3x
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V3x ) @ V2b2 ) )
& ( V1b1
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V2b2 ) @ V3x ) ) @ V4y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Edominates @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V4y ) ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V3x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Edominates__DIFF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ bool ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( arr @ A_27a @ bool ) )
=> ! [V3i: $i] :
( ( mem @ V3i @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Edominates @ A_27a @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3i ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3i ) @ V1x ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3i ) @ V2y ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Edominates @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1x ) @ V3i ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V2y ) @ V3i ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INSERT__SUB__BAG__E,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V0e ) @ V1b ) ) @ V2c ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__IN @ A_27a ) @ V0e ) @ V2c ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V1b ) @ V2c ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Ebdominates__BAG__DIFF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3i: $i] :
( ( mem @ V3i @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Edominates @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V1x ) ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V2y ) ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V3i ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V3i ) @ V1x ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V3i ) @ V2y ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Edominates @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1x ) @ V3i ) ) ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V2y ) @ V3i ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INTER__SUB__BAG,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INTER @ A_27a ) @ V0b1 ) @ V1b2 ) ) @ V0b1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INTER @ A_27a ) @ V0b1 ) @ V1b2 ) ) @ V1b2 ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__INTER__FINITE,axiom,
! [A_27a: del,V0b1: $i] :
( ( mem @ V0b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b2: $i] :
( ( mem @ V1b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b1 ) )
| ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b2 ) ) )
=> ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INTER @ A_27a ) @ V0b1 ) @ V1b2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt__dominates__thm2,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) ) )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V1b1 ) @ V2b2 ) )
<=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V2b2 ) )
& ? [V3x: $i] :
( ( mem @ V3x @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ? [V4y: $i] :
( ( mem @ V4y @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( V3x
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V3x ) @ V2b2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DISJOINT @ A_27a ) @ V3x ) @ V4y ) )
& ( V1b1
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V2b2 ) @ V3x ) ) @ V4y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebag_2Edominates @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V4y ) ) @ ( ap @ ( c_2Ebag_2ESET__OF__BAG @ A_27a ) @ V3x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__DIFF__INSERT__SUB__BAG,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0c ) @ V1b ) )
=> ( ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e ) @ V1b ) ) @ V0c )
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V1b ) @ V0c ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt__INSERT__CANCEL,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ V2a ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ V3b ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V2a ) @ V3b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt__UNION__RCANCEL__I,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V1a ) @ V2b ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V3c ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1a ) @ V3c ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V2b ) @ V3c ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt__UNION__RCANCEL,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1a ) @ V2c ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V3b ) @ V2c ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V1a ) @ V3b ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V2c ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt__UNION__LCANCEL__I,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V1a ) @ V2b ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V3c ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V3c ) @ V1a ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V3c ) @ V2b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt__UNION__LCANCEL,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1c ) @ V2a ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1c ) @ V3b ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V2a ) @ V3b ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1c ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt__UNION__CANCEL__EQN,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1b1: $i] :
( ( mem @ V1b1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b2: $i] :
( ( mem @ V2b2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V1b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V1b1 ) @ V2b2 ) ) )
<=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V2b2 ) )
& ( V2b2
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ A_27a @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ A_27a ) @ V0R ) ) @ V1b1 ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__UNION @ A_27a ) @ V2b2 ) @ V1b1 ) ) )
<=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V1b1 ) )
& ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V2b2 ) )
& ( V2b2
!= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2Emlt__UNION__EMPTY__EQN,axiom,
$true ).
thf(conj_thm_2Ebag_2ESUB__BAG__SING,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ V0b ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) )
<=> ( ( V0b
= ( c_2Ebag_2EEMPTY__BAG @ A_27a ) )
| ( V0b
= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V1e ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2ESUB__BAG__DIFF__simple,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebag_2ESUB__BAG @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__DIFF @ A_27a ) @ V0b ) @ V1c ) ) @ V0b ) ) ) ) ).
thf(conj_thm_2Ebag_2EmltLT__SING0,axiom,
! [V0b: $i] :
( ( mem @ V0b @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Ebag_2Emlt1 @ ty_2Enum_2Enum ) @ c_2Eprim__rec_2E_3C ) ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Ebag_2EEMPTY__BAG @ ty_2Enum_2Enum ) ) ) @ V0b ) )
<=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ ty_2Enum_2Enum ) @ V0b ) )
& ( V0b
!= ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Ebag_2EEMPTY__BAG @ ty_2Enum_2Enum ) ) )
& ( V0b
!= ( c_2Ebag_2EEMPTY__BAG @ ty_2Enum_2Enum ) ) ) ) ) ).
thf(ax_thm_2Ebag_2Ebag__size__def,axiom,
! [A_27a: del,V0eltsize: $i] :
( ( mem @ V0eltsize @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Ebag_2Ebag__size @ A_27a ) @ V0eltsize ) @ V1b ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap
@ ( ap @ ( c_2Ebag_2EITBAG @ A_27a @ ty_2Enum_2Enum )
@ ( lam @ A_27a
@ ^ [V2e: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V3acc: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ V0eltsize @ V2e ) ) ) @ V3acc ) ) ) )
@ V1b )
@ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ebag_2EBAG__SIZE__EMPTY,axiom,
! [A_27a: del,V0eltsize: $i] :
( ( mem @ V0eltsize @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Ebag_2Ebag__size @ A_27a ) @ V0eltsize ) @ ( c_2Ebag_2EEMPTY__BAG @ A_27a ) ) )
= fo__c_2Enum_2E0 ) ) ).
thf(conj_thm_2Ebag_2EBAG__SIZE__INSERT,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1eltsize: $i] :
( ( mem @ V1eltsize @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( p @ ( ap @ ( c_2Ebag_2EFINITE__BAG @ A_27a ) @ V0b ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Ebag_2Ebag__size @ A_27a ) @ V1eltsize ) @ ( ap @ ( ap @ ( c_2Ebag_2EBAG__INSERT @ A_27a ) @ V2e ) @ V0b ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ V1eltsize @ V2e ) ) ) @ ( ap @ ( ap @ ( c_2Ebag_2Ebag__size @ A_27a ) @ V1eltsize ) @ V0b ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------