TPTP Problem File: ITP024^3.p
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%------------------------------------------------------------------------------
% File : ITP024^3 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 syntactic export of thm_2Ereal__topology_2ECLOSED__UNION__COMPACT__SUBSETS.p, bushy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : thm_2Ereal__topology_2ECLOSED__UNION__COMPACT__SUBSETS.p [Gau19]
% : HL411501^3.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 153 ( 37 unt; 56 typ; 0 def)
% Number of atoms : 370 ( 118 equ; 49 cnn)
% Maximal formula atoms : 41 ( 3 avg)
% Number of connectives : 1300 ( 49 ~; 35 |; 122 &; 972 @)
% ( 71 <=>; 51 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 7 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 155 ( 155 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 53 usr; 5 con; 0-5 aty)
% Number of variables : 335 ( 18 ^; 286 !; 16 ?; 335 :)
% ( 15 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Earithmetic_2E_3E,type,
c_2Earithmetic_2E_3E: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3E_3D,type,
c_2Earithmetic_2E_3E_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EBIGUNION,type,
c_2Epred__set_2EBIGUNION:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ereal__topology_2EClosed,type,
c_2Ereal__topology_2EClosed: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2EDist,type,
c_2Ereal__topology_2EDist: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Earithmetic_2EEVEN,type,
c_2Earithmetic_2EEVEN: tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2EEXP,type,
c_2Earithmetic_2EEXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EODD,type,
c_2Earithmetic_2EODD: tyop_2Enum_2Enum > $o ).
thf(c_2Eprim__rec_2EPRE,type,
c_2Eprim__rec_2EPRE: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ereal_2Eabs,type,
c_2Ereal_2Eabs: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal__topology_2Eball,type,
c_2Ereal__topology_2Eball: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal__topology_2Ebounded__def,type,
c_2Ereal__topology_2Ebounded__def: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Ereal__topology_2Ecball,type,
c_2Ereal__topology_2Ecball: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal__topology_2Ecompact,type,
c_2Ereal__topology_2Ecompact: ( tyop_2Erealax_2Ereal > $o ) > $o ).
thf(c_2Enumeral_2EiDUB,type,
c_2Enumeral_2EiDUB: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Enumeral_2EiZ,type,
c_2Enumeral_2EiZ: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Enumeral_2EiiSUC,type,
c_2Enumeral_2EiiSUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__ge,type,
c_2Ereal_2Ereal__ge: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Ereal__lt,type,
c_2Erealax_2Ereal__lt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2Ereal__lte,type,
c_2Ereal_2Ereal__lte: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Ereal__mul,type,
c_2Erealax_2Ereal__mul: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__neg,type,
c_2Erealax_2Ereal__neg: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__of__num,type,
c_2Ereal_2Ereal__of__num: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Earithmetic_2EZERO__LESS__EQ,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ c_2Enum_2E0 @ V0n ) ).
thf(thm_2Earithmetic_2ENOT__LESS__EQUAL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) )
<=> ( c_2Eprim__rec_2E_3C @ V1n @ V0m ) ) ).
thf(thm_2Earithmetic_2EADD__EQ__0,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2B @ V0m @ V1n )
= c_2Enum_2E0 )
<=> ( ( V0m = c_2Enum_2E0 )
& ( V1n = c_2Enum_2E0 ) ) ) ).
thf(thm_2Earithmetic_2ELE,axiom,
( ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0n @ c_2Enum_2E0 )
<=> ( V0n = c_2Enum_2E0 ) )
& ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ ( c_2Enum_2ESUC @ V2n ) )
<=> ( ( V1m
= ( c_2Enum_2ESUC @ V2n ) )
| ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n ) ) ) ) ).
thf(thm_2Ebool_2EIN__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Ebool_2EIN @ A_27a )
= ( ^ [V0x: A_27a,V1f: A_27a > $o] : ( V1f @ V0x ) ) ) ).
thf(thm_2Ebool_2ETRUTH,axiom,
c_2Ebool_2ET ).
thf(thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1: $o,V1t2: $o] :
( ( V0t1
=> V1t2 )
=> ( ( V1t2
=> V0t1 )
=> ( V0t1 = V1t2 ) ) ) ).
thf(thm_2Ebool_2EFORALL__SIMP,axiom,
! [A_27a: $tType,V0t: $o] :
( ! [V1x: A_27a] : V0t
<=> V0t ) ).
thf(thm_2Ebool_2EIMP__F,axiom,
! [V0t: $o] :
( ( V0t
=> c_2Ebool_2EF )
=> ( (~) @ V0t ) ) ).
thf(thm_2Ebool_2EF__IMP,axiom,
! [V0t: $o] :
( ( (~) @ V0t )
=> ( V0t
=> c_2Ebool_2EF ) ) ).
thf(thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET
& V0t )
<=> V0t )
& ( ( V0t
& c_2Ebool_2ET )
<=> V0t )
& ( ( c_2Ebool_2EF
& V0t )
<=> c_2Ebool_2EF )
& ( ( V0t
& c_2Ebool_2EF )
<=> c_2Ebool_2EF )
& ( ( V0t
& V0t )
<=> V0t ) ) ).
thf(thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET
=> V0t )
<=> V0t )
& ( ( V0t
=> c_2Ebool_2ET )
<=> c_2Ebool_2ET )
& ( ( c_2Ebool_2EF
=> V0t )
<=> c_2Ebool_2ET )
& ( ( V0t
=> V0t )
<=> c_2Ebool_2ET )
& ( ( V0t
=> c_2Ebool_2EF )
<=> ( (~) @ V0t ) ) ) ).
thf(thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: $o] :
( ( (~) @ ( (~) @ V0t ) )
<=> V0t )
& ( ( (~) @ c_2Ebool_2ET )
<=> c_2Ebool_2EF )
& ( ( (~) @ c_2Ebool_2EF )
<=> c_2Ebool_2ET ) ) ).
thf(thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( V0x = V0x )
<=> c_2Ebool_2ET ) ).
thf(thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
( ( V0x = V1y )
<=> ( V1y = V0x ) ) ).
thf(thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET = V0t )
<=> V0t )
& ( ( V0t = c_2Ebool_2ET )
<=> V0t )
& ( ( c_2Ebool_2EF = V0t )
<=> ( (~) @ V0t ) )
& ( ( V0t = c_2Ebool_2EF )
<=> ( (~) @ V0t ) ) ) ).
thf(thm_2Ebool_2ENOT__EXISTS__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( (~)
@ ? [V1x: A_27a] : ( V0P @ V1x ) )
<=> ! [V2x: A_27a] : ( (~) @ ( V0P @ V2x ) ) ) ).
thf(thm_2Ebool_2ELEFT__OR__EXISTS__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
( ( ? [V2x: A_27a] : ( V0P @ V2x )
| V1Q )
<=> ? [V3x: A_27a] :
( ( V0P @ V3x )
| V1Q ) ) ).
thf(thm_2Ebool_2ELEFT__EXISTS__AND__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
( ? [V2x: A_27a] :
( ( V0P @ V2x )
& V1Q )
<=> ( ? [V3x: A_27a] : ( V0P @ V3x )
& V1Q ) ) ).
thf(thm_2Ebool_2ERIGHT__EXISTS__AND__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
( ? [V2x: A_27a] :
( V0P
& ( V1Q @ V2x ) )
<=> ( V0P
& ? [V3x: A_27a] : ( V1Q @ V3x ) ) ) ).
thf(thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A: $o,V1B: $o] :
( ( ( (~)
@ ( V0A
& V1B ) )
<=> ( ( (~) @ V0A )
| ( (~) @ V1B ) ) )
& ( ( (~)
@ ( V0A
| V1B ) )
<=> ( ( (~) @ V0A )
& ( (~) @ V1B ) ) ) ) ).
thf(thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1: $o,V1t2: $o,V2t3: $o] :
( ( V0t1
=> ( V1t2
=> V2t3 ) )
<=> ( ( V0t1
& V1t2 )
=> V2t3 ) ) ).
thf(thm_2Ebool_2EIMP__CONG,axiom,
! [V0x: $o,V1x_27: $o,V2y: $o,V3y_27: $o] :
( ( ( V0x = V1x_27 )
& ( V1x_27
=> ( V2y = V3y_27 ) ) )
=> ( ( V0x
=> V2y )
<=> ( V1x_27
=> V3y_27 ) ) ) ).
thf(thm_2Ebool_2EUNWIND__THM2,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1a: A_27a] :
( ? [V2x: A_27a] :
( ( V2x = V1a )
& ( V0P @ V2x ) )
<=> ( V0P @ V1a ) ) ).
thf(thm_2Ebool_2ESKOLEM__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
( ! [V1x: A_27a] :
? [V2y: A_27b] : ( V0P @ V1x @ V2y )
<=> ? [V3f: A_27a > A_27b] :
! [V4x: A_27a] : ( V0P @ V4x @ ( V3f @ V4x ) ) ) ).
thf(thm_2Ecardinal_2ECONJ__EQ__IMP,axiom,
! [V0r: $o,V1p: $o,V2q: $o] :
( ( ( V1p
& V2q )
=> V0r )
<=> ( V1p
=> ( V2q
=> V0r ) ) ) ).
thf(thm_2Eiterate_2ESIMP__REAL__ARCH,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
? [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ V1n ) ) ).
thf(thm_2Enumeral_2Enumeral__distrib,axiom,
( ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ c_2Enum_2E0 @ V0n )
= V0n )
& ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ V1n @ c_2Enum_2E0 )
= V1n )
& ! [V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ V2n ) @ ( c_2Earithmetic_2ENUMERAL @ V3m ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Enumeral_2EiZ @ ( c_2Earithmetic_2E_2B @ V2n @ V3m ) ) ) )
& ! [V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ c_2Enum_2E0 @ V4n )
= c_2Enum_2E0 )
& ! [V5n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ V5n @ c_2Enum_2E0 )
= c_2Enum_2E0 )
& ! [V6n: tyop_2Enum_2Enum,V7m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ V6n ) @ ( c_2Earithmetic_2ENUMERAL @ V7m ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2E_2A @ V6n @ V7m ) ) )
& ! [V8n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ c_2Enum_2E0 @ V8n )
= c_2Enum_2E0 )
& ! [V9n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ V9n @ c_2Enum_2E0 )
= V9n )
& ! [V10n: tyop_2Enum_2Enum,V11m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ V10n ) @ ( c_2Earithmetic_2ENUMERAL @ V11m ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2E_2D @ V10n @ V11m ) ) )
& ! [V12n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ c_2Enum_2E0 @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V12n ) ) )
= c_2Enum_2E0 )
& ! [V13n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ c_2Enum_2E0 @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V13n ) ) )
= c_2Enum_2E0 )
& ! [V14n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ V14n @ c_2Enum_2E0 )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V15n: tyop_2Enum_2Enum,V16m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ V15n ) @ ( c_2Earithmetic_2ENUMERAL @ V16m ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EEXP @ V15n @ V16m ) ) )
& ( ( c_2Enum_2ESUC @ c_2Enum_2E0 )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V17n: tyop_2Enum_2Enum] :
( ( c_2Enum_2ESUC @ ( c_2Earithmetic_2ENUMERAL @ V17n ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Enum_2ESUC @ V17n ) ) )
& ( ( c_2Eprim__rec_2EPRE @ c_2Enum_2E0 )
= c_2Enum_2E0 )
& ! [V18n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2EPRE @ ( c_2Earithmetic_2ENUMERAL @ V18n ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Eprim__rec_2EPRE @ V18n ) ) )
& ! [V19n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2ENUMERAL @ V19n )
= c_2Enum_2E0 )
<=> ( V19n = c_2Earithmetic_2EZERO ) )
& ! [V20n: tyop_2Enum_2Enum] :
( ( c_2Enum_2E0
= ( c_2Earithmetic_2ENUMERAL @ V20n ) )
<=> ( V20n = c_2Earithmetic_2EZERO ) )
& ! [V21n: tyop_2Enum_2Enum,V22m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2ENUMERAL @ V21n )
= ( c_2Earithmetic_2ENUMERAL @ V22m ) )
<=> ( V21n = V22m ) )
& ! [V23n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V23n @ c_2Enum_2E0 )
= c_2Ebool_2EF )
& ! [V24n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Earithmetic_2ENUMERAL @ V24n ) )
= ( c_2Eprim__rec_2E_3C @ c_2Earithmetic_2EZERO @ V24n ) )
& ! [V25n: tyop_2Enum_2Enum,V26m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ V25n ) @ ( c_2Earithmetic_2ENUMERAL @ V26m ) )
= ( c_2Eprim__rec_2E_3C @ V25n @ V26m ) )
& ! [V27n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E @ c_2Enum_2E0 @ V27n )
= c_2Ebool_2EF )
& ! [V28n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E @ ( c_2Earithmetic_2ENUMERAL @ V28n ) @ c_2Enum_2E0 )
= ( c_2Eprim__rec_2E_3C @ c_2Earithmetic_2EZERO @ V28n ) )
& ! [V29n: tyop_2Enum_2Enum,V30m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E @ ( c_2Earithmetic_2ENUMERAL @ V29n ) @ ( c_2Earithmetic_2ENUMERAL @ V30m ) )
= ( c_2Eprim__rec_2E_3C @ V30m @ V29n ) )
& ! [V31n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ c_2Enum_2E0 @ V31n )
= c_2Ebool_2ET )
& ! [V32n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ V32n ) @ c_2Enum_2E0 )
= ( c_2Earithmetic_2E_3C_3D @ V32n @ c_2Earithmetic_2EZERO ) )
& ! [V33n: tyop_2Enum_2Enum,V34m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ V33n ) @ ( c_2Earithmetic_2ENUMERAL @ V34m ) )
= ( c_2Earithmetic_2E_3C_3D @ V33n @ V34m ) )
& ! [V35n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V35n @ c_2Enum_2E0 )
= c_2Ebool_2ET )
& ! [V36n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ c_2Enum_2E0 @ V36n )
<=> ( V36n = c_2Enum_2E0 ) )
& ! [V37n: tyop_2Enum_2Enum,V38m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ ( c_2Earithmetic_2ENUMERAL @ V37n ) @ ( c_2Earithmetic_2ENUMERAL @ V38m ) )
= ( c_2Earithmetic_2E_3C_3D @ V38m @ V37n ) )
& ! [V39n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EODD @ ( c_2Earithmetic_2ENUMERAL @ V39n ) )
= ( c_2Earithmetic_2EODD @ V39n ) )
& ! [V40n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ ( c_2Earithmetic_2ENUMERAL @ V40n ) )
= ( c_2Earithmetic_2EEVEN @ V40n ) )
& ( (~) @ ( c_2Earithmetic_2EODD @ c_2Enum_2E0 ) )
& ( c_2Earithmetic_2EEVEN @ c_2Enum_2E0 ) ) ).
thf(thm_2Enumeral_2Enumeral__add,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Enumeral_2EiZ @ ( c_2Earithmetic_2E_2B @ c_2Earithmetic_2EZERO @ V0n ) )
= V0n )
& ( ( c_2Enumeral_2EiZ @ ( c_2Earithmetic_2E_2B @ V0n @ c_2Earithmetic_2EZERO ) )
= V0n )
& ( ( c_2Enumeral_2EiZ @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EBIT1 @ V0n ) @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) )
= ( c_2Earithmetic_2EBIT2 @ ( c_2Enumeral_2EiZ @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) )
& ( ( c_2Enumeral_2EiZ @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EBIT1 @ V0n ) @ ( c_2Earithmetic_2EBIT2 @ V1m ) ) )
= ( c_2Earithmetic_2EBIT1 @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) )
& ( ( c_2Enumeral_2EiZ @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EBIT2 @ V0n ) @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) )
= ( c_2Earithmetic_2EBIT1 @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) )
& ( ( c_2Enumeral_2EiZ @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EBIT2 @ V0n ) @ ( c_2Earithmetic_2EBIT2 @ V1m ) ) )
= ( c_2Earithmetic_2EBIT2 @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) )
& ( ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ c_2Earithmetic_2EZERO @ V0n ) )
= ( c_2Enum_2ESUC @ V0n ) )
& ( ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V0n @ c_2Earithmetic_2EZERO ) )
= ( c_2Enum_2ESUC @ V0n ) )
& ( ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EBIT1 @ V0n ) @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) )
= ( c_2Earithmetic_2EBIT1 @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) )
& ( ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EBIT1 @ V0n ) @ ( c_2Earithmetic_2EBIT2 @ V1m ) ) )
= ( c_2Earithmetic_2EBIT2 @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) )
& ( ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EBIT2 @ V0n ) @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) )
= ( c_2Earithmetic_2EBIT2 @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) )
& ( ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EBIT2 @ V0n ) @ ( c_2Earithmetic_2EBIT2 @ V1m ) ) )
= ( c_2Earithmetic_2EBIT1 @ ( c_2Enumeral_2EiiSUC @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) )
& ( ( c_2Enumeral_2EiiSUC @ ( c_2Earithmetic_2E_2B @ c_2Earithmetic_2EZERO @ V0n ) )
= ( c_2Enumeral_2EiiSUC @ V0n ) )
& ( ( c_2Enumeral_2EiiSUC @ ( c_2Earithmetic_2E_2B @ V0n @ c_2Earithmetic_2EZERO ) )
= ( c_2Enumeral_2EiiSUC @ V0n ) )
& ( ( c_2Enumeral_2EiiSUC @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EBIT1 @ V0n ) @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) )
= ( c_2Earithmetic_2EBIT2 @ ( c_2Enum_2ESUC @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) )
& ( ( c_2Enumeral_2EiiSUC @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EBIT1 @ V0n ) @ ( c_2Earithmetic_2EBIT2 @ V1m ) ) )
= ( c_2Earithmetic_2EBIT1 @ ( c_2Enumeral_2EiiSUC @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) )
& ( ( c_2Enumeral_2EiiSUC @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EBIT2 @ V0n ) @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) )
= ( c_2Earithmetic_2EBIT1 @ ( c_2Enumeral_2EiiSUC @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) )
& ( ( c_2Enumeral_2EiiSUC @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EBIT2 @ V0n ) @ ( c_2Earithmetic_2EBIT2 @ V1m ) ) )
= ( c_2Earithmetic_2EBIT2 @ ( c_2Enumeral_2EiiSUC @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) ) ) ).
thf(thm_2Enumeral_2EiDUB__removal,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( ( c_2Enumeral_2EiDUB @ ( c_2Earithmetic_2EBIT1 @ V0n ) )
= ( c_2Earithmetic_2EBIT2 @ ( c_2Enumeral_2EiDUB @ V0n ) ) )
& ( ( c_2Enumeral_2EiDUB @ ( c_2Earithmetic_2EBIT2 @ V0n ) )
= ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ V0n ) ) )
& ( ( c_2Enumeral_2EiDUB @ c_2Earithmetic_2EZERO )
= c_2Earithmetic_2EZERO ) ) ).
thf(thm_2Enumeral_2Enumeral__mult,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2A @ c_2Earithmetic_2EZERO @ V0n )
= c_2Earithmetic_2EZERO )
& ( ( c_2Earithmetic_2E_2A @ V0n @ c_2Earithmetic_2EZERO )
= c_2Earithmetic_2EZERO )
& ( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2EBIT1 @ V0n ) @ V1m )
= ( c_2Enumeral_2EiZ @ ( c_2Earithmetic_2E_2B @ ( c_2Enumeral_2EiDUB @ ( c_2Earithmetic_2E_2A @ V0n @ V1m ) ) @ V1m ) ) )
& ( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2EBIT2 @ V0n ) @ V1m )
= ( c_2Enumeral_2EiDUB @ ( c_2Enumeral_2EiZ @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ V0n @ V1m ) @ V1m ) ) ) ) ) ).
thf(thm_2Epair_2ECLOSED__PAIR__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b,V2a: A_27a,V3b: A_27b] :
( ( ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V1y )
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2a @ V3b ) )
<=> ( ( V0x = V2a )
& ( V1y = V3b ) ) ) ).
thf(thm_2Epred__set_2EEXTENSION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( V0s = V1t )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
= ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ).
thf(thm_2Epred__set_2EGSPECIFICATION,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ),V1v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1v @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b @ V0f ) )
<=> ? [V2x: A_27b] :
( ( c_2Epair_2E_2C @ A_27a @ $o @ V1v @ c_2Ebool_2ET )
= ( V0f @ V2x ) ) ) ).
thf(thm_2Epred__set_2EIN__UNIV,axiom,
! [A_27a: $tType,V0x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(thm_2Epred__set_2ESUBSET__DEF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
=> ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__TRANS,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V2u ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V2u ) ) ).
thf(thm_2Epred__set_2EIN__INTER,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ).
thf(thm_2Epred__set_2EINTER__SUBSET,axiom,
! [A_27a: $tType] :
( ! [V0s: A_27a > $o,V1t: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) @ V0s )
& ! [V2s: A_27a > $o,V3t: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V3t @ V2s ) @ V2s ) ) ).
thf(thm_2Epred__set_2ESUBSET__INTER,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ ( c_2Epred__set_2EINTER @ A_27a @ V1t @ V2u ) )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V2u ) ) ) ).
thf(thm_2Ereal_2EREAL__ADD__SYM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V0x @ V1y )
= ( c_2Erealax_2Ereal__add @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__ADD__ASSOC,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Erealax_2Ereal__add @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ V2z ) ) ).
thf(thm_2Ereal_2EREAL__ADD__LID,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
= V0x ) ).
thf(thm_2Ereal_2EREAL__ADD__LINV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__neg @ V0x ) @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__LT__TRANS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
& ( c_2Erealax_2Ereal__lt @ V1y @ V2z ) )
=> ( c_2Erealax_2Ereal__lt @ V0x @ V2z ) ) ).
thf(thm_2Ereal_2EREAL__MUL__LID,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x )
= V0x ) ).
thf(thm_2Ereal_2Ereal__ge,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__ge @ V0x @ V1y )
= ( c_2Ereal_2Ereal__lte @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__ADD__RID,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__ADD__RINV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Erealax_2Ereal__neg @ V0x ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__MUL__RID,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__NEG__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__neg @ V0x ) @ ( c_2Erealax_2Ereal__neg @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__MUL__LZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__LT__LADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__add @ V0x @ V2z ) )
= ( c_2Erealax_2Ereal__lt @ V1y @ V2z ) ) ).
thf(thm_2Ereal_2EREAL__LTE__TRANS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
& ( c_2Ereal_2Ereal__lte @ V1y @ V2z ) )
=> ( c_2Erealax_2Ereal__lt @ V0x @ V2z ) ) ).
thf(thm_2Ereal_2EREAL__LET__TRANS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
& ( c_2Erealax_2Ereal__lt @ V1y @ V2z ) )
=> ( c_2Erealax_2Ereal__lt @ V0x @ V2z ) ) ).
thf(thm_2Ereal_2EREAL__LE__TRANS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
& ( c_2Ereal_2Ereal__lte @ V1y @ V2z ) )
=> ( c_2Ereal_2Ereal__lte @ V0x @ V2z ) ) ).
thf(thm_2Ereal_2EREAL__LE__ANTISYM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
& ( c_2Ereal_2Ereal__lte @ V1y @ V0x ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ereal_2EREAL__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ V0m ) @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) ) ) ).
thf(thm_2Ereal_2EREAL__MUL__RNEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Ereal__neg @ V1y ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__MUL__LNEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__neg @ V0x ) @ V1y )
= ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2Ereal__lt,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V1x @ V0y )
<=> ( (~) @ ( c_2Ereal_2Ereal__lte @ V0y @ V1x ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__LADD__IMP,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V1y @ V2z )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__add @ V0x @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__LNEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ V0x ) @ V1y )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__NEG2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ V0x ) @ ( c_2Erealax_2Ereal__neg @ V1y ) )
= ( c_2Ereal_2Ereal__lte @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__NEG__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__neg @ V0x ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__LE__RNEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Erealax_2Ereal__neg @ V1y ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EREAL__ADD__RDISTRIB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ V2z )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__OF__NUM__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ V0m ) @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) ) ) ).
thf(thm_2Ereal_2EREAL__OF__NUM__LE,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ V0m ) @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
= ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) ) ).
thf(thm_2Ereal_2EREAL__OF__NUM__MUL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V0m ) @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) ) ) ).
thf(thm_2Ereal__topology_2EREAL__OF__NUM__GE,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__ge @ ( c_2Ereal_2Ereal__of__num @ V0m ) @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
= ( c_2Earithmetic_2E_3E_3D @ V0m @ V1n ) ) ).
thf(thm_2Ereal__topology_2EDIST__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V0x ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal__topology_2EIN__CBALL__0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e ) ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V0x ) @ V1e ) ) ).
thf(thm_2Ereal__topology_2EBOUNDED__SUBSET__CBALL,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ebounded__def @ V0s )
=> ? [V2r: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2r )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1x @ V2r ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ESUBSET__BALLS,axiom,
( ! [V0a: tyop_2Erealax_2Ereal,V1a_27: tyop_2Erealax_2Ereal,V2r: tyop_2Erealax_2Ereal,V3r_27: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V2r ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a_27 @ V3r_27 ) ) )
<=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1a_27 ) ) @ V2r ) @ V3r_27 )
| ( c_2Ereal_2Ereal__lte @ V2r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
& ! [V4a: tyop_2Erealax_2Ereal,V5a_27: tyop_2Erealax_2Ereal,V6r: tyop_2Erealax_2Ereal,V7r_27: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V6r ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V5a_27 @ V7r_27 ) ) )
<=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V4a @ V5a_27 ) ) @ V6r ) @ V7r_27 )
| ( c_2Ereal_2Ereal__lte @ V6r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
& ! [V8a: tyop_2Erealax_2Ereal,V9a_27: tyop_2Erealax_2Ereal,V10r: tyop_2Erealax_2Ereal,V11r_27: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V10r ) ) @ ( c_2Ereal__topology_2Eball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V9a_27 @ V11r_27 ) ) )
<=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V9a_27 ) ) @ V10r ) @ V11r_27 )
| ( c_2Erealax_2Ereal__lt @ V10r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
& ! [V12a: tyop_2Erealax_2Ereal,V13a_27: tyop_2Erealax_2Ereal,V14r: tyop_2Erealax_2Ereal,V15r_27: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V12a @ V14r ) ) @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V13a_27 @ V15r_27 ) ) )
<=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal__topology_2EDist @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V12a @ V13a_27 ) ) @ V14r ) @ V15r_27 )
| ( c_2Erealax_2Ereal__lt @ V14r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__IMP__BOUNDED,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2Ecompact @ V0s )
=> ( c_2Ereal__topology_2Ebounded__def @ V0s ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__INTER__COMPACT,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2EClosed @ V0s )
& ( c_2Ereal__topology_2Ecompact @ V1t ) )
=> ( c_2Ereal__topology_2Ecompact @ ( c_2Epred__set_2EINTER @ tyop_2Erealax_2Ereal @ V0s @ V1t ) ) ) ).
thf(thm_2Ereal__topology_2ECOMPACT__CBALL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1e: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecompact @ ( c_2Ereal__topology_2Ecball @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1e ) ) ) ).
thf(thm_2Ereal__topology_2EBIGUNION__GSPEC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,A_27h: $tType,A_27i: $tType] :
( ! [V0P: A_27a > $o,V1f: A_27a > A_27b > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27b
@ ( c_2Epred__set_2EGSPEC @ ( A_27b > $o ) @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ ( A_27b > $o ) @ $o @ ( V1f @ V2x ) @ ( V0P @ V2x ) ) ) )
= ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V3a: A_27b] :
( c_2Epair_2E_2C @ A_27b @ $o @ V3a
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V0P @ V4x ) @ ( c_2Ebool_2EIN @ A_27b @ V3a @ ( V1f @ V4x ) ) ) ) ) ) )
& ! [V5P: A_27c > A_27d > $o,V6f: A_27c > A_27d > A_27e > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27e
@ ( c_2Epred__set_2EGSPEC @ ( A_27e > $o ) @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d )
@ ( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ ( tyop_2Epair_2Eprod @ ( A_27e > $o ) @ $o )
@ ^ [V7x: A_27c,V8y: A_27d] : ( c_2Epair_2E_2C @ ( A_27e > $o ) @ $o @ ( V6f @ V7x @ V8y ) @ ( V5P @ V7x @ V8y ) ) ) ) )
= ( c_2Epred__set_2EGSPEC @ A_27e @ A_27e
@ ^ [V9a: A_27e] :
( c_2Epair_2E_2C @ A_27e @ $o @ V9a
@ ( c_2Ebool_2E_3F @ A_27c
@ ^ [V10x: A_27c] :
( c_2Ebool_2E_3F @ A_27d
@ ^ [V11y: A_27d] : ( c_2Ebool_2E_2F_5C @ ( V5P @ V10x @ V11y ) @ ( c_2Ebool_2EIN @ A_27e @ V9a @ ( V6f @ V10x @ V11y ) ) ) ) ) ) ) )
& ! [V12P: A_27f > A_27g > A_27h > $o,V13f: A_27f > A_27g > A_27h > A_27i > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27i
@ ( c_2Epred__set_2EGSPEC @ ( A_27i > $o ) @ ( tyop_2Epair_2Eprod @ A_27f @ ( tyop_2Epair_2Eprod @ A_27g @ A_27h ) )
@ ( c_2Epair_2EUNCURRY @ A_27f @ ( tyop_2Epair_2Eprod @ A_27g @ A_27h ) @ ( tyop_2Epair_2Eprod @ ( A_27i > $o ) @ $o )
@ ^ [V14x: A_27f] :
( c_2Epair_2EUNCURRY @ A_27g @ A_27h @ ( tyop_2Epair_2Eprod @ ( A_27i > $o ) @ $o )
@ ^ [V15y: A_27g,V16z: A_27h] : ( c_2Epair_2E_2C @ ( A_27i > $o ) @ $o @ ( V13f @ V14x @ V15y @ V16z ) @ ( V12P @ V14x @ V15y @ V16z ) ) ) ) ) )
= ( c_2Epred__set_2EGSPEC @ A_27i @ A_27i
@ ^ [V17a: A_27i] :
( c_2Epair_2E_2C @ A_27i @ $o @ V17a
@ ( c_2Ebool_2E_3F @ A_27f
@ ^ [V18x: A_27f] :
( c_2Ebool_2E_3F @ A_27g
@ ^ [V19y: A_27g] :
( c_2Ebool_2E_3F @ A_27h
@ ^ [V20z: A_27h] : ( c_2Ebool_2E_2F_5C @ ( V12P @ V18x @ V19y @ V20z ) @ ( c_2Ebool_2EIN @ A_27i @ V17a @ ( V13f @ V18x @ V19y @ V20z ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Esat_2ENOT__NOT,axiom,
! [V0t: $o] :
( ( (~) @ ( (~) @ V0t ) )
<=> V0t ) ).
thf(thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A: $o] :
( V0A
=> ( ( (~) @ V0A )
=> c_2Ebool_2EF ) ) ).
thf(thm_2Esat_2EOR__DUAL2,axiom,
! [V0B: $o,V1A: $o] :
( ( ( (~)
@ ( V1A
| V0B ) )
=> c_2Ebool_2EF )
<=> ( ( V1A
=> c_2Ebool_2EF )
=> ( ( (~) @ V0B )
=> c_2Ebool_2EF ) ) ) ).
thf(thm_2Esat_2EOR__DUAL3,axiom,
! [V0B: $o,V1A: $o] :
( ( ( (~)
@ ( ( (~) @ V1A )
| V0B ) )
=> c_2Ebool_2EF )
<=> ( V1A
=> ( ( (~) @ V0B )
=> c_2Ebool_2EF ) ) ) ).
thf(thm_2Esat_2EAND__INV2,axiom,
! [V0A: $o] :
( ( ( (~) @ V0A )
=> c_2Ebool_2EF )
=> ( ( V0A
=> c_2Ebool_2EF )
=> c_2Ebool_2EF ) ) ).
thf(thm_2Esat_2Edc__eq,axiom,
! [V0r: $o,V1q: $o,V2p: $o] :
( ( V2p
<=> ( V1q = V0r ) )
<=> ( ( V2p
| V1q
| V0r )
& ( V2p
| ( (~) @ V0r )
| ( (~) @ V1q ) )
& ( V1q
| ( (~) @ V0r )
| ( (~) @ V2p ) )
& ( V0r
| ( (~) @ V1q )
| ( (~) @ V2p ) ) ) ) ).
thf(thm_2Esat_2Edc__conj,axiom,
! [V0r: $o,V1q: $o,V2p: $o] :
( ( V2p
<=> ( V1q
& V0r ) )
<=> ( ( V2p
| ( (~) @ V1q )
| ( (~) @ V0r ) )
& ( V1q
| ( (~) @ V2p ) )
& ( V0r
| ( (~) @ V2p ) ) ) ) ).
thf(thm_2Esat_2Edc__disj,axiom,
! [V0r: $o,V1q: $o,V2p: $o] :
( ( V2p
<=> ( V1q
| V0r ) )
<=> ( ( V2p
| ( (~) @ V1q ) )
& ( V2p
| ( (~) @ V0r ) )
& ( V1q
| V0r
| ( (~) @ V2p ) ) ) ) ).
thf(thm_2Esat_2Edc__imp,axiom,
! [V0r: $o,V1q: $o,V2p: $o] :
( ( V2p
<=> ( V1q
=> V0r ) )
<=> ( ( V2p
| V1q )
& ( V2p
| ( (~) @ V0r ) )
& ( ( (~) @ V1q )
| V0r
| ( (~) @ V2p ) ) ) ) ).
thf(thm_2Esat_2Edc__neg,axiom,
! [V0q: $o,V1p: $o] :
( ( V1p
<=> ( (~) @ V0q ) )
<=> ( ( V1p
| V0q )
& ( ( (~) @ V0q )
| ( (~) @ V1p ) ) ) ) ).
thf(thm_2Ereal__topology_2ECLOSED__UNION__COMPACT__SUBSETS,conjecture,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal__topology_2EClosed @ V0s )
=> ? [V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > $o] :
( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2Ecompact @ ( V1f @ V2n ) )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V1f @ V3n ) @ V0s )
& ! [V4n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ ( V1f @ V4n ) @ ( V1f @ ( c_2Earithmetic_2E_2B @ V4n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( c_2Epred__set_2EBIGUNION @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Erealax_2Ereal > $o ) @ tyop_2Enum_2Enum
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( tyop_2Erealax_2Ereal > $o ) @ $o @ ( V1f @ V5n ) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5n @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
= V0s )
& ! [V6k: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Ereal__topology_2Ecompact @ V6k )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V6k @ V0s ) )
=> ? [V7N: tyop_2Enum_2Enum] :
! [V8n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V8n @ V7N )
=> ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V6k @ ( V1f @ V8n ) ) ) ) ) ) ).
%------------------------------------------------------------------------------