ITP001 Axioms: ITP147^7.ax
%------------------------------------------------------------------------------
% File : ITP147^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : product.ax [Gau19]
% : HL4147^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 144 ( 33 unt; 54 typ; 0 def)
% Number of atoms : 344 ( 102 equ; 3 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 1742 ( 3 ~; 1 |; 62 &;1581 @)
% ( 11 <=>; 84 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 8 avg;1581 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 267 ( 267 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 51 usr; 3 con; 0-6 aty)
% Number of variables : 407 ( 38 ^ 336 !; 5 ?; 407 :)
% ( 28 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(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_2Eiterate_2E_2E_2E,type,
c_2Eiterate_2E_2E_2E: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Ereal_2E_2F,type,
c_2Ereal_2E_2F: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
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_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2ECARD,type,
c_2Epred__set_2ECARD:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Epred__set_2EDELETE,type,
c_2Epred__set_2EDELETE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epred__set_2EDISJOINT,type,
c_2Epred__set_2EDISJOINT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EEXP,type,
c_2Earithmetic_2EEXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EFACT,type,
c_2Earithmetic_2EFACT: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $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_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: 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_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ereal_2Eabs,type,
c_2Ereal_2Eabs: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Einv,type,
c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Eiterate_2Eiterate,type,
c_2Eiterate_2Eiterate:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27b > A_27b ) > ( A_27a > $o ) > ( A_27a > A_27b ) > A_27b ) ).
thf(c_2Eproduct_2Enproduct,type,
c_2Eproduct_2Enproduct:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Ereal_2Epow,type,
c_2Ereal_2Epow: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Eproduct_2Eproduct,type,
c_2Eproduct_2Eproduct:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
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_2Eiterate_2Esupport,type,
c_2Eiterate_2Esupport:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27b > A_27b ) > ( A_27a > A_27b ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Eproduct_2Eproduct,axiom,
! [A_27a: $tType] :
( ( c_2Eproduct_2Eproduct @ A_27a )
= ( c_2Eiterate_2Eiterate @ A_27a @ tyop_2Erealax_2Ereal @ c_2Erealax_2Ereal__mul ) ) ).
thf(thm_2Eproduct_2Enproduct,axiom,
! [A_27a: $tType] :
( ( c_2Eproduct_2Enproduct @ A_27a )
= ( c_2Eiterate_2Eiterate @ A_27a @ tyop_2Enum_2Enum @ c_2Earithmetic_2E_2A ) ) ).
thf(thm_2Eproduct_2EPRODUCT__DELTA,axiom,
! [A_27a: $tType,V0b: tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2a: A_27a] :
( ( c_2Eproduct_2Eproduct @ A_27a @ V1s
@ ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V2a ) @ V0b @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s ) @ V0b @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__DELETE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2a: A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s ) )
=> ( ( c_2Erealax_2Ereal__mul @ ( V0f @ V2a ) @ ( c_2Eproduct_2Eproduct @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V2a ) @ V0f ) )
= ( c_2Eproduct_2Eproduct @ A_27a @ V1s @ V0f ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__PAIR,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0f )
= ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0f @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V3i ) ) @ ( V0f @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V3i ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__SUPERSET,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1u: A_27a > $o,V2v: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1u @ V2v )
& ! [V3x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2v )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1u ) ) )
=> ( ( V0f @ V3x )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( ( c_2Eproduct_2Eproduct @ A_27a @ V2v @ V0f )
= ( c_2Eproduct_2Eproduct @ A_27a @ V1u @ V0f ) ) ) ).
thf(thm_2Eproduct_2EREAL__OF__NUM__NPRODUCT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Ereal_2Ereal__of__num @ ( c_2Eproduct_2Enproduct @ A_27a @ V1s @ V0f ) )
= ( c_2Eproduct_2Eproduct @ A_27a @ V1s
@ ^ [V2x: A_27a] : ( c_2Ereal_2Ereal__of__num @ ( V0f @ V2x ) ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__CLAUSES__RIGHT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
& ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n ) )
=> ( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Erealax_2Ereal__mul @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0f ) @ ( V0f @ V2n ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__CLAUSES__LEFT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
=> ( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Erealax_2Ereal__mul @ ( V0f @ V1m ) @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V1m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__CLOSED,axiom,
! [A_27a: $tType,V0P: tyop_2Erealax_2Ereal > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( V0P @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( V0P @ V3x )
& ( V0P @ V4y ) )
=> ( V0P @ ( c_2Erealax_2Ereal__mul @ V3x @ V4y ) ) )
& ! [V5a: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5a @ V2s )
=> ( V0P @ ( V1f @ V5a ) ) ) )
=> ( V0P @ ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ V1f ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__ABS,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Eproduct_2Eproduct @ A_27a @ V1s
@ ^ [V2x: A_27a] : ( c_2Ereal_2Eabs @ ( V0f @ V2x ) ) )
= ( c_2Ereal_2Eabs @ ( c_2Eproduct_2Eproduct @ A_27a @ V1s @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__LE__1,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V2x ) )
& ( c_2Ereal_2Ereal__lte @ ( V0f @ V2x ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eproduct_2Eproduct @ A_27a @ V1s @ V0f ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__ONE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Eproduct_2Eproduct @ A_27a @ V0s
@ ^ [V1n: A_27a] : ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__DIV__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V4x: tyop_2Enum_2Enum] : ( c_2Ereal_2E_2F @ ( V0f @ V4x ) @ ( V1g @ V4x ) ) )
= ( c_2Ereal_2E_2F @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__DIV,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( c_2Eproduct_2Eproduct @ A_27a @ V2s
@ ^ [V3x: A_27a] : ( c_2Ereal_2E_2F @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) )
= ( c_2Ereal_2E_2F @ ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ V0f ) @ ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ V1g ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__INV,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Eproduct_2Eproduct @ A_27a @ V1s
@ ^ [V2x: A_27a] : ( c_2Erealax_2Einv @ ( V0f @ V2x ) ) )
= ( c_2Erealax_2Einv @ ( c_2Eproduct_2Eproduct @ A_27a @ V1s @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__NEG__NUMSEG__1,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1n )
@ ^ [V2i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( V0f @ V2i ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V1n ) @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1n ) @ V0f ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__NEG__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( V0f @ V3i ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1m ) ) @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__NEG,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Eproduct_2Eproduct @ A_27a @ V1s
@ ^ [V2i: A_27a] : ( c_2Erealax_2Ereal__neg @ ( V0f @ V2i ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) ) @ ( c_2Eproduct_2Eproduct @ A_27a @ V1s @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__CONST__NUMSEG__1,axiom,
! [V0c: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1n )
@ ^ [V2x: tyop_2Enum_2Enum] : V0c )
= ( c_2Ereal_2Epow @ V0c @ V1n ) ) ).
thf(thm_2Eproduct_2EPRODUCT__CONST__NUMSEG,axiom,
! [V0c: tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3x: tyop_2Enum_2Enum] : V0c )
= ( c_2Ereal_2Epow @ V0c @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1m ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__CONST,axiom,
! [A_27a: $tType,V0c: tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Eproduct_2Eproduct @ A_27a @ V1s
@ ^ [V2x: A_27a] : V0c )
= ( c_2Ereal_2Epow @ V0c @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__MUL__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V4x: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0f @ V4x ) @ ( V1g @ V4x ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__MUL,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( c_2Eproduct_2Eproduct @ A_27a @ V2s
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ V0f ) @ ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ V1g ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__MUL__GEN,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V1g @ V4x ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
=> ( ( c_2Eproduct_2Eproduct @ A_27a @ V2s
@ ^ [V5x: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V0f @ V5x ) @ ( V1g @ V5x ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ V0f ) @ ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ V1g ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__EQ__1__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ! [V3i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V3i )
& ( c_2Earithmetic_2E_3C_3D @ V3i @ V2n ) )
=> ( ( V0f @ V3i )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__EQ__1,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( ( V0f @ V2x )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( c_2Eproduct_2Eproduct @ A_27a @ V1s @ V0f )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__LE__NUMSEG,axiom,
! [V0g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ! [V4i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V4i )
& ( c_2Earithmetic_2E_3C_3D @ V4i @ V3n ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V4i ) )
& ( c_2Ereal_2Ereal__lte @ ( V1f @ V4i ) @ ( V0g @ V4i ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1f ) @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0g ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__LE,axiom,
! [A_27a: $tType,V0g: A_27a > tyop_2Erealax_2Ereal,V1f: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) )
& ( c_2Ereal_2Ereal__lte @ ( V1f @ V3x ) @ ( V0g @ V3x ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ V1f ) @ ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ V0g ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__EQ__0__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ? [V3x: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V3x )
& ( c_2Earithmetic_2E_3C_3D @ V3x @ V2n )
& ( ( V0f @ V3x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__EQ__0,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( ( c_2Eproduct_2Eproduct @ A_27a @ V1s @ V0f )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ? [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( ( V0f @ V2x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__EQ__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ! [V4i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V4i )
& ( c_2Earithmetic_2E_3C_3D @ V4i @ V3n ) )
=> ( ( V0f @ V4i )
= ( V1g @ V4i ) ) )
=> ( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f )
= ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__EQ,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( ( V0f @ V3x )
= ( V1g @ V3x ) ) )
=> ( ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ V0f )
= ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__CLAUSES__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ! [V1m: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ c_2Enum_2E0 ) @ V0f )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1m @ c_2Enum_2E0 ) @ ( V0f @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ ( c_2Enum_2ESUC @ V3n ) ) @ V0f )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2E_3C_3D @ V2m @ ( c_2Enum_2ESUC @ V3n ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) @ ( V0f @ ( c_2Enum_2ESUC @ V3n ) ) ) @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__SING__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1n @ V1n ) @ V0f )
= ( V0f @ V1n ) ) ).
thf(thm_2Eproduct_2EPRODUCT__SING,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1x: A_27a] :
( ( c_2Eproduct_2Eproduct @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0f )
= ( V0f @ V1x ) ) ).
thf(thm_2Eproduct_2EPRODUCT__OFFSET,axiom,
! [V0n: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3p: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V2m @ V3p ) @ ( c_2Earithmetic_2E_2B @ V0n @ V3p ) ) @ V1f )
= ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V0n )
@ ^ [V4i: tyop_2Enum_2Enum] : ( V1f @ ( c_2Earithmetic_2E_2B @ V4i @ V3p ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__POS__LT__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ! [V3x: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V3x )
& ( c_2Earithmetic_2E_3C_3D @ V3x @ V2n ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V3x ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__POS__LT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V2x ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eproduct_2Eproduct @ A_27a @ V1s @ V0f ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__POS__LE__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ! [V3x: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V3x )
& ( c_2Earithmetic_2E_3C_3D @ V3x @ V2n ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V3x ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__POS__LE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V2x ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eproduct_2Eproduct @ A_27a @ V1s @ V0f ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__ADD__SPLIT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ ( c_2Earithmetic_2E_2B @ V2n @ V3p ) ) @ V0f )
= ( c_2Erealax_2Ereal__mul @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f ) @ ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2E_2B @ V2n @ V3p ) ) @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27b > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V2s )
& ( ( V0f @ V3x )
= ( V0f @ V4y ) ) )
=> ( V3x = V4y ) )
=> ( ( c_2Eproduct_2Eproduct @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s ) @ V1g )
= ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Erealax_2Ereal @ A_27b @ V1g @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__UNION,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V2t )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V1s @ V2t ) )
=> ( ( c_2Eproduct_2Eproduct @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ V0f )
= ( c_2Erealax_2Ereal__mul @ ( c_2Eproduct_2Eproduct @ A_27a @ V1s @ V0f ) @ ( c_2Eproduct_2Eproduct @ A_27a @ V2t @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2EPRODUCT__SUPPORT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Eproduct_2Eproduct @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ tyop_2Erealax_2Ereal @ c_2Erealax_2Ereal__mul @ V0f @ V1s ) @ V0f )
= ( c_2Eproduct_2Eproduct @ A_27a @ V1s @ V0f ) ) ).
thf(thm_2Eproduct_2EPRODUCT__CLAUSES,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Eproduct_2Eproduct @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0f )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V1x: A_27b,V2f: A_27b > tyop_2Erealax_2Ereal,V3s: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ V3s )
=> ( ( c_2Eproduct_2Eproduct @ A_27b @ ( c_2Epred__set_2EINSERT @ A_27b @ V1x @ V3s ) @ V2f )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ A_27b @ V1x @ V3s ) @ ( c_2Eproduct_2Eproduct @ A_27b @ V3s @ V2f ) @ ( c_2Erealax_2Ereal__mul @ ( V2f @ V1x ) @ ( c_2Eproduct_2Eproduct @ A_27b @ V3s @ V2f ) ) ) ) ) ) ).
thf(thm_2Eproduct_2EREAL__ADD__AC,axiom,
! [V0p: tyop_2Erealax_2Ereal,V1n: tyop_2Erealax_2Ereal,V2m: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__add @ V2m @ V1n )
= ( c_2Erealax_2Ereal__add @ V1n @ V2m ) )
& ( ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__add @ V2m @ V1n ) @ V0p )
= ( c_2Erealax_2Ereal__add @ V2m @ ( c_2Erealax_2Ereal__add @ V1n @ V0p ) ) )
& ( ( c_2Erealax_2Ereal__add @ V2m @ ( c_2Erealax_2Ereal__add @ V1n @ V0p ) )
= ( c_2Erealax_2Ereal__add @ V1n @ ( c_2Erealax_2Ereal__add @ V2m @ V0p ) ) ) ) ).
thf(thm_2Eproduct_2EMULT__AC,axiom,
! [V0p: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_2A @ V2m @ V1n )
= ( c_2Earithmetic_2E_2A @ V1n @ V2m ) )
& ( ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2E_2A @ V2m @ V1n ) @ V0p )
= ( c_2Earithmetic_2E_2A @ V2m @ ( c_2Earithmetic_2E_2A @ V1n @ V0p ) ) )
& ( ( c_2Earithmetic_2E_2A @ V2m @ ( c_2Earithmetic_2E_2A @ V1n @ V0p ) )
= ( c_2Earithmetic_2E_2A @ V1n @ ( c_2Earithmetic_2E_2A @ V2m @ V0p ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__CLAUSES,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27a > tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Enproduct @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0f )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V1x: A_27b,V2f: A_27b > tyop_2Enum_2Enum,V3s: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ V3s )
=> ( ( c_2Eproduct_2Enproduct @ A_27b @ ( c_2Epred__set_2EINSERT @ A_27b @ V1x @ V3s ) @ V2f )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ A_27b @ V1x @ V3s ) @ ( c_2Eproduct_2Enproduct @ A_27b @ V3s @ V2f ) @ ( c_2Earithmetic_2E_2A @ ( V2f @ V1x ) @ ( c_2Eproduct_2Enproduct @ A_27b @ V3s @ V2f ) ) ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__SUPPORT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Eproduct_2Enproduct @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ tyop_2Enum_2Enum @ c_2Earithmetic_2E_2A @ V0f @ V1s ) @ V0f )
= ( c_2Eproduct_2Enproduct @ A_27a @ V1s @ V0f ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__UNION,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V2t )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V1s @ V2t ) )
=> ( ( c_2Eproduct_2Enproduct @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ V0f )
= ( c_2Earithmetic_2E_2A @ ( c_2Eproduct_2Enproduct @ A_27a @ V1s @ V0f ) @ ( c_2Eproduct_2Enproduct @ A_27a @ V2t @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27b > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V2s )
& ( ( V0f @ V3x )
= ( V0f @ V4y ) ) )
=> ( V3x = V4y ) )
=> ( ( c_2Eproduct_2Enproduct @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s ) @ V1g )
= ( c_2Eproduct_2Enproduct @ A_27a @ V2s @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Enum_2Enum @ A_27b @ V1g @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__ADD__SPLIT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ ( c_2Earithmetic_2E_2B @ V2n @ V3p ) ) @ V0f )
= ( c_2Earithmetic_2E_2A @ ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f ) @ ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2E_2B @ V2n @ V3p ) ) @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__POS__LT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( V0f @ V2x ) ) ) )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Eproduct_2Enproduct @ A_27a @ V1s @ V0f ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__POS__LT__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ! [V3x: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V3x )
& ( c_2Earithmetic_2E_3C_3D @ V3x @ V2n ) )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( V0f @ V3x ) ) )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__OFFSET,axiom,
! [V0n: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3p: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V2m @ V3p ) @ ( c_2Earithmetic_2E_2B @ V0n @ V3p ) ) @ V1f )
= ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V0n )
@ ^ [V4i: tyop_2Enum_2Enum] : ( V1f @ ( c_2Earithmetic_2E_2B @ V4i @ V3p ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__SING,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1x: A_27a] :
( ( c_2Eproduct_2Enproduct @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0f )
= ( V0f @ V1x ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__SING__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1n @ V1n ) @ V0f )
= ( V0f @ V1n ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__CLAUSES__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V1m: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ c_2Enum_2E0 ) @ V0f )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1m @ c_2Enum_2E0 ) @ ( V0f @ c_2Enum_2E0 ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ ( c_2Enum_2ESUC @ V3n ) ) @ V0f )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_3C_3D @ V2m @ ( c_2Enum_2ESUC @ V3n ) ) @ ( c_2Earithmetic_2E_2A @ ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) @ ( V0f @ ( c_2Enum_2ESUC @ V3n ) ) ) @ ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__EQ,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1g: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( ( V0f @ V3x )
= ( V1g @ V3x ) ) )
=> ( ( c_2Eproduct_2Enproduct @ A_27a @ V2s @ V0f )
= ( c_2Eproduct_2Enproduct @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__EQ__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1g: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ! [V4i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V4i )
& ( c_2Earithmetic_2E_3C_3D @ V4i @ V3n ) )
=> ( ( V0f @ V4i )
= ( V1g @ V4i ) ) )
=> ( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f )
= ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__EQ__0,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( ( c_2Eproduct_2Enproduct @ A_27a @ V1s @ V0f )
= c_2Enum_2E0 )
<=> ? [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( ( V0f @ V2x )
= c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__EQ__0__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= c_2Enum_2E0 )
<=> ? [V3x: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V3x )
& ( c_2Earithmetic_2E_3C_3D @ V3x @ V2n )
& ( ( V0f @ V3x )
= c_2Enum_2E0 ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__LE,axiom,
! [A_27a: $tType,V0g: A_27a > tyop_2Enum_2Enum,V1f: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( ( c_2Earithmetic_2E_3C_3D @ c_2Enum_2E0 @ ( V1f @ V3x ) )
& ( c_2Earithmetic_2E_3C_3D @ ( V1f @ V3x ) @ ( V0g @ V3x ) ) ) ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Eproduct_2Enproduct @ A_27a @ V2s @ V1f ) @ ( c_2Eproduct_2Enproduct @ A_27a @ V2s @ V0g ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__LE__NUMSEG,axiom,
! [V0g: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ! [V4i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V4i )
& ( c_2Earithmetic_2E_3C_3D @ V4i @ V3n ) )
=> ( ( c_2Earithmetic_2E_3C_3D @ c_2Enum_2E0 @ ( V1f @ V4i ) )
& ( c_2Earithmetic_2E_3C_3D @ ( V1f @ V4i ) @ ( V0g @ V4i ) ) ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1f ) @ ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0g ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__EQ__1,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( ( V0f @ V2x )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( ( c_2Eproduct_2Enproduct @ A_27a @ V1s @ V0f )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__EQ__1__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ! [V3i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V3i )
& ( c_2Earithmetic_2E_3C_3D @ V3i @ V2n ) )
=> ( ( V0f @ V3i )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__MUL__GEN,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1g: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ ( V0f @ V3x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ ( V1g @ V4x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
=> ( ( c_2Eproduct_2Enproduct @ A_27a @ V2s
@ ^ [V5x: A_27a] : ( c_2Earithmetic_2E_2A @ ( V0f @ V5x ) @ ( V1g @ V5x ) ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Eproduct_2Enproduct @ A_27a @ V2s @ V0f ) @ ( c_2Eproduct_2Enproduct @ A_27a @ V2s @ V1g ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__MUL,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1g: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( c_2Eproduct_2Enproduct @ A_27a @ V2s
@ ^ [V3x: A_27a] : ( c_2Earithmetic_2E_2A @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Eproduct_2Enproduct @ A_27a @ V2s @ V0f ) @ ( c_2Eproduct_2Enproduct @ A_27a @ V2s @ V1g ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__MUL__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1g: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V4x: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A @ ( V0f @ V4x ) @ ( V1g @ V4x ) ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) @ ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__CONST,axiom,
! [A_27a: $tType,V0c: tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Eproduct_2Enproduct @ A_27a @ V1s
@ ^ [V2x: A_27a] : V0c )
= ( c_2Earithmetic_2EEXP @ V0c @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__CONST__NUMSEG,axiom,
! [V0c: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3x: tyop_2Enum_2Enum] : V0c )
= ( c_2Earithmetic_2EEXP @ V0c @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1m ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__CONST__NUMSEG__1,axiom,
! [V0c: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1n )
@ ^ [V2x: tyop_2Enum_2Enum] : V0c )
= ( c_2Earithmetic_2EEXP @ V0c @ V1n ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__ONE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Eproduct_2Enproduct @ A_27a @ V0s
@ ^ [V1n: A_27a] : ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__CLOSED,axiom,
! [A_27a: $tType,V0P: tyop_2Enum_2Enum > $o,V1f: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( ( V0P @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V3x: tyop_2Enum_2Enum,V4y: tyop_2Enum_2Enum] :
( ( ( V0P @ V3x )
& ( V0P @ V4y ) )
=> ( V0P @ ( c_2Earithmetic_2E_2A @ V3x @ V4y ) ) )
& ! [V5a: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5a @ V2s )
=> ( V0P @ ( V1f @ V5a ) ) ) )
=> ( V0P @ ( c_2Eproduct_2Enproduct @ A_27a @ V2s @ V1f ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__CLAUSES__LEFT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
=> ( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Earithmetic_2E_2A @ ( V0f @ V1m ) @ ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V1m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) @ V0f ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__CLAUSES__RIGHT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
& ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n ) )
=> ( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Earithmetic_2E_2A @ ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0f ) @ ( V0f @ V2n ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__SUPERSET,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1u: A_27a > $o,V2v: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1u @ V2v )
& ! [V3x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2v )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1u ) ) )
=> ( ( V0f @ V3x )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( c_2Eproduct_2Enproduct @ A_27a @ V2v @ V0f )
= ( c_2Eproduct_2Enproduct @ A_27a @ V1u @ V0f ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__PAIR,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0f )
= ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A @ ( V0f @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V3i ) ) @ ( V0f @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V3i ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__DELETE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2a: A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s ) )
=> ( ( c_2Earithmetic_2E_2A @ ( V0f @ V2a ) @ ( c_2Eproduct_2Enproduct @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V2a ) @ V0f ) )
= ( c_2Eproduct_2Enproduct @ A_27a @ V1s @ V0f ) ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__FACT,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eproduct_2Enproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n )
@ ^ [V1m: tyop_2Enum_2Enum] : V1m )
= ( c_2Earithmetic_2EFACT @ V0n ) ) ).
thf(thm_2Eproduct_2ENPRODUCT__DELTA,axiom,
! [A_27a: $tType,V0b: tyop_2Enum_2Enum,V1s: A_27a > $o,V2a: A_27a] :
( ( c_2Eproduct_2Enproduct @ A_27a @ V1s
@ ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V2a ) @ V0b @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s ) @ V0b @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Eproduct_2Eth,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( ( V0f @ V3x )
= ( V1g @ V3x ) ) )
=> ( ( c_2Eproduct_2Eproduct @ A_27a @ V2s
@ ^ [V4i: A_27a] : ( V0f @ V4i ) )
= ( c_2Eproduct_2Eproduct @ A_27a @ V2s @ V1g ) ) )
& ! [V5f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V6g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V7a: tyop_2Enum_2Enum,V8b: tyop_2Enum_2Enum] :
( ! [V9i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V7a @ V9i )
& ( c_2Earithmetic_2E_3C_3D @ V9i @ V8b ) )
=> ( ( V5f @ V9i )
= ( V6g @ V9i ) ) )
=> ( ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V7a @ V8b )
@ ^ [V10i: tyop_2Enum_2Enum] : ( V5f @ V10i ) )
= ( c_2Eproduct_2Eproduct @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V7a @ V8b ) @ V6g ) ) )
& ! [V11f: A_27b > tyop_2Erealax_2Ereal,V12g: A_27b > tyop_2Erealax_2Ereal,V13p: A_27b > $o] :
( ! [V14x: A_27b] :
( ( V13p @ V14x )
=> ( ( V11f @ V14x )
= ( V12g @ V14x ) ) )
=> ( ( c_2Eproduct_2Eproduct @ A_27b
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V15y: A_27b] : ( c_2Epair_2E_2C @ A_27b @ $o @ V15y @ ( V13p @ V15y ) ) )
@ ^ [V16i: A_27b] : ( V11f @ V16i ) )
= ( c_2Eproduct_2Eproduct @ A_27b
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V17y: A_27b] : ( c_2Epair_2E_2C @ A_27b @ $o @ V17y @ ( V13p @ V17y ) ) )
@ V12g ) ) ) ) ).
%------------------------------------------------------------------------------