ITP001 Axioms: ITP118^7.ax
%------------------------------------------------------------------------------
% File : ITP118^7 : 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 : real.ax [Gau19]
% : HL4118^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 528 ( 228 unt; 62 typ; 0 def)
% Number of atoms : 1138 ( 431 equ; 58 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 5368 ( 58 ~; 20 |; 172 &;4843 @)
% ( 74 <=>; 201 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg;4843 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 162 ( 162 >; 0 *; 0 +; 0 <<)
% Number of symbols : 61 ( 59 usr; 7 con; 0-4 aty)
% Number of variables : 1132 ( 31 ^1028 !; 60 ?;1132 :)
% ( 13 !>; 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_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_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_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
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_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Earithmetic_2EDIV,type,
c_2Earithmetic_2EDIV: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_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_2Ewhile_2ELEAST,type,
c_2Ewhile_2ELEAST: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ELET,type,
c_2Ebool_2ELET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ereal_2ENUM__CEILING,type,
c_2Ereal_2ENUM__CEILING: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum ).
thf(c_2Ereal_2ENUM__FLOOR,type,
c_2Ereal_2ENUM__FLOOR: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EODD,type,
c_2Earithmetic_2EODD: tyop_2Enum_2Enum > $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_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_2Einf,type,
c_2Ereal_2Einf: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Einv,type,
c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Emax,type,
c_2Ereal_2Emax: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Emin,type,
c_2Ereal_2Emin: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Epos,type,
c_2Ereal_2Epos: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Epow,type,
c_2Ereal_2Epow: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__0,type,
c_2Erealax_2Ereal__0: tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__1,type,
c_2Erealax_2Ereal__1: tyop_2Erealax_2Ereal ).
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_2Ereal_2Ereal__gt,type,
c_2Ereal_2Ereal__gt: 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_2Ereal_2Ereal__sub,type,
c_2Ereal_2Ereal__sub: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Esum,type,
c_2Ereal_2Esum: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Esup,type,
c_2Ereal_2Esup: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal ).
thf(c_2Emarker_2Eunint,type,
c_2Emarker_2Eunint:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
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_2Ereal_2Ereal__of__num,axiom,
( ( ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
= c_2Erealax_2Ereal__0 )
& ! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__of__num @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ c_2Erealax_2Ereal__1 ) ) ) ).
thf(thm_2Ereal_2Ereal__sub,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ V0x @ V1y )
= ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Erealax_2Ereal__neg @ V1y ) ) ) ).
thf(thm_2Ereal_2Ereal__lte,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
<=> ( (~) @ ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ) ).
thf(thm_2Ereal_2Ereal__gt,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__gt @ V0x @ V1y )
= ( c_2Erealax_2Ereal__lt @ V1y @ 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__div,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2E_2F @ V0x @ V1y )
= ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Einv @ V1y ) ) ) ).
thf(thm_2Ereal_2Eabs,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ V0x )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) @ V0x @ ( c_2Erealax_2Ereal__neg @ V0x ) ) ) ).
thf(thm_2Ereal_2Epow,axiom,
( ! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epow @ V0x @ c_2Enum_2E0 )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V1x: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ V1x @ ( c_2Enum_2ESUC @ V2n ) )
= ( c_2Erealax_2Ereal__mul @ V1x @ ( c_2Ereal_2Epow @ V1x @ V2n ) ) ) ) ).
thf(thm_2Ereal_2Esup,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal_2Esup @ V0P )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V1s: tyop_2Erealax_2Ereal] :
( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V2y: tyop_2Erealax_2Ereal] :
( c_2Emin_2E_3D @ $o
@ ( c_2Ebool_2E_3F @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( V0P @ V3x ) @ ( c_2Erealax_2Ereal__lt @ V2y @ V3x ) ) )
@ ( c_2Erealax_2Ereal__lt @ V2y @ V1s ) ) ) ) ) ).
thf(thm_2Ereal_2Epos__def,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epos @ V0x )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2Emin__def,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Emin @ V0x @ V1y )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) @ V0x @ V1y ) ) ).
thf(thm_2Ereal_2Emax__def,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Emax @ V0x @ V1y )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2Einf__def,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal_2Einf @ V0p )
= ( c_2Erealax_2Ereal__neg
@ ( c_2Ereal_2Esup
@ ^ [V1r: tyop_2Erealax_2Ereal] : ( V0p @ ( c_2Erealax_2Ereal__neg @ V1r ) ) ) ) ) ).
thf(thm_2Ereal_2ENUM__FLOOR__def,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2ENUM__FLOOR @ V0x )
= ( c_2Ewhile_2ELEAST
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__gt @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2B @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0x ) ) ) ).
thf(thm_2Ereal_2ENUM__CEILING__def,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2ENUM__CEILING @ V0x )
= ( c_2Ewhile_2ELEAST
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ V1n ) ) ) ) ).
thf(thm_2Ereal_2EREAL__0,axiom,
( c_2Erealax_2Ereal__0
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__1,axiom,
( c_2Erealax_2Ereal__1
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ereal_2EREAL__10,axiom,
( (~)
@ ( ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
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__COMM,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__LDISTRIB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Ereal__add @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__TOTAL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( V0x = V1y )
| ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
| ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LT__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( (~) @ ( c_2Erealax_2Ereal__lt @ V0x @ V0x ) ) ).
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__LT__IADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V1y @ V2z )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__add @ V0x @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__SUP__ALLPOS,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ! [V1x: tyop_2Erealax_2Ereal] :
( ( V0P @ V1x )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x ) )
& ? [V2x: tyop_2Erealax_2Ereal] : ( V0P @ V2x )
& ? [V3z: tyop_2Erealax_2Ereal] :
! [V4x: tyop_2Erealax_2Ereal] :
( ( V0P @ V4x )
=> ( c_2Erealax_2Ereal__lt @ V4x @ V3z ) ) )
=> ? [V5s: tyop_2Erealax_2Ereal] :
! [V6y: tyop_2Erealax_2Ereal] :
( ? [V7x: tyop_2Erealax_2Ereal] :
( ( V0P @ V7x )
& ( c_2Erealax_2Ereal__lt @ V6y @ V7x ) )
<=> ( c_2Erealax_2Ereal__lt @ V6y @ V5s ) ) ) ).
thf(thm_2Ereal_2EREAL__MUL__SYM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Erealax_2Ereal__mul @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__MUL__COMM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Erealax_2Ereal__mul @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__MUL__ASSOC,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ 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__MUL__LINV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0x ) @ V0x )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__INV__0,axiom,
( ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
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__MUL__RINV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Einv @ V0x ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__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__EQ__LADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__add @ V0x @ V1y )
= ( c_2Erealax_2Ereal__add @ V0x @ V2z ) )
<=> ( V1y = V2z ) ) ).
thf(thm_2Ereal_2EREAL__EQ__RADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__add @ V0x @ V2z )
= ( c_2Erealax_2Ereal__add @ V1y @ V2z ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ereal_2EREAL__ADD__LID__UNIQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__add @ V0x @ V1y )
= V1y )
<=> ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EREAL__ADD__RID__UNIQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__add @ V0x @ V1y )
= V0x )
<=> ( V1y
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EREAL__LNEG__UNIQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__add @ V0x @ V1y )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( V0x
= ( c_2Erealax_2Ereal__neg @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__RNEG__UNIQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__add @ V0x @ V1y )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( V1y
= ( c_2Erealax_2Ereal__neg @ 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__MUL__RZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__NEG__LMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__neg @ V0x ) @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__NEG__RMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Ereal__neg @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__NEGNEG,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__neg @ V0x ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__NEG__MUL2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__neg @ V0x ) @ ( c_2Erealax_2Ereal__neg @ V1y ) )
= ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__ENTIRE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( V1y
= ( 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__LT__RADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V0x @ V2z ) @ ( c_2Erealax_2Ereal__add @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__lt @ V0x @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__NOT__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~) @ ( c_2Erealax_2Ereal__lt @ V0x @ V1y ) )
<=> ( c_2Ereal_2Ereal__lte @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LT__ANTISYM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( (~)
@ ( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
& ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__GT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
=> ( (~) @ ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__NOT__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~) @ ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) )
<=> ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LE__TOTAL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
| ( c_2Ereal_2Ereal__lte @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LET__TOTAL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
| ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LTE__TOTAL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
| ( c_2Ereal_2Ereal__lte @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LE__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ V0x @ V0x ) ).
thf(thm_2Ereal_2EREAL__LE__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
<=> ( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
| ( V0x = V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
<=> ( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
& ( (~) @ ( V0x = V1y ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__IMP__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
=> ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ).
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__LET__ANTISYM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( (~)
@ ( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
& ( c_2Ereal_2Ereal__lte @ V1y @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__LTE__ANTISYM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( (~)
@ ( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
& ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__LTE__ANTSYM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( (~)
@ ( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
& ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__NEG__LT0,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ V0x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__NEG__GT0,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__neg @ V0x ) )
= ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EREAL__NEG__LE0,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ V0x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__NEG__GE0,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__neg @ V0x ) )
= ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__NEGTOTAL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
| ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__neg @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__NEGTOTAL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
| ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__neg @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__SQUARE,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LE__01,axiom,
c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__01,axiom,
c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__LADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__add @ V0x @ V2z ) )
= ( c_2Ereal_2Ereal__lte @ V1y @ V2z ) ) ).
thf(thm_2Ereal_2EREAL__LE__RADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ V0x @ V2z ) @ ( c_2Erealax_2Ereal__add @ V1y @ V2z ) )
= ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__LT__ADD2,axiom,
! [V0w: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V0w @ V1x )
& ( c_2Erealax_2Ereal__lt @ V2y @ V3z ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V0w @ V2y ) @ ( c_2Erealax_2Ereal__add @ V1x @ V3z ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__ADD2,axiom,
! [V0w: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0w @ V1x )
& ( c_2Ereal_2Ereal__lte @ V2y @ V3z ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ V0w @ V2y ) @ ( c_2Erealax_2Ereal__add @ V1x @ V3z ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__ADDNEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V1y @ ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Erealax_2Ereal__neg @ V2z ) ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V1y @ V2z ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LT__ADDNEG2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Erealax_2Ereal__neg @ V1y ) ) @ V2z )
= ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Erealax_2Ereal__add @ V2z @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__ADD1,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
=> ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Erealax_2Ereal__add @ V1y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) @ V1y )
= V0x ) ).
thf(thm_2Ereal_2EREAL__SUB__ADD2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V1y @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__SUB__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ V0x @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__SUB__0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__sub @ V0x @ V1y )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ereal_2EREAL__LE__DOUBLE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__add @ V0x @ V0x ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LE__NEGL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ V0x ) @ V0x )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LE__NEGR,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Erealax_2Ereal__neg @ V0x ) )
= ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EREAL__NEG__EQ0,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__neg @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EREAL__NEG__0,axiom,
( ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__NEG__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) )
= ( c_2Ereal_2Ereal__sub @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__SUB__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__SUB__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) )
= ( c_2Ereal_2Ereal__lte @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__ADD__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ V0x )
= V1y ) ).
thf(thm_2Ereal_2EREAL__EQ__LMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) )
<=> ( ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( V1y = V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__EQ__RMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__mul @ V0x @ V2z )
= ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) )
<=> ( ( V2z
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( V0x = V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB__LDISTRIB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Ereal_2Ereal__sub @ V1y @ V2z ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB__RDISTRIB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) @ V2z )
= ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__NEG__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__neg @ V0x )
= V1y )
<=> ( V0x
= ( c_2Erealax_2Ereal__neg @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__NEG__MINUS1,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg @ V0x )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__INV__NZ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( (~)
@ ( ( c_2Erealax_2Einv @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal_2EREAL__INVINV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Einv @ ( c_2Erealax_2Einv @ V0x ) )
= V0x ) ) ).
thf(thm_2Ereal_2EREAL__LT__IMP__NE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
=> ( (~) @ ( V0x = V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__INV__POS,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Einv @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__LMUL__0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__RMUL__0,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__LMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) )
= ( c_2Erealax_2Ereal__lt @ V1y @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__RMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2z )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__lt @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__RMUL__IMP,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2z ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__LMUL__IMP,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V1y @ V2z )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__LINV__UNIQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( V0x
= ( c_2Erealax_2Einv @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__RINV__UNIQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( V1y
= ( c_2Erealax_2Einv @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__INV__INV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Einv @ ( c_2Erealax_2Einv @ V0x ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__INV__EQ__0,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Einv @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EREAL__NEG__INV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Einv @ V0x ) )
= ( c_2Erealax_2Einv @ ( c_2Erealax_2Ereal__neg @ V0x ) ) ) ) ).
thf(thm_2Ereal_2EREAL__INV__1OVER,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Einv @ V0x )
= ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LT__INV__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Einv @ V0x ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LE__INV__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Einv @ V0x ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LE__INV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Einv @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__ADDR,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__LE__ADDL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V1y @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LT__ADDR,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__LT__ADDL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V1y @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__of__num @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__POS,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) ).
thf(thm_2Ereal_2EREAL__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__LT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ V0m ) @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
= ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Ereal_2EREAL__INJ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Ereal_2Ereal__of__num @ V0m )
= ( c_2Ereal_2Ereal__of__num @ V1n ) )
<=> ( V0m = V1n ) ) ).
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,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_2EREAL__INV1,axiom,
( ( c_2Erealax_2Einv @ ( 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_2Ereal_2EREAL__OVER1,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__DIV__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal_2E_2F @ V0x @ V0x )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__DIV__LZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__LT__NZ,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( (~)
@ ( ( c_2Ereal_2Ereal__of__num @ V0n )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
<=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) ) ).
thf(thm_2Ereal_2EREAL__NZ__IMP__LT,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( (~) @ ( V0n = c_2Enum_2E0 ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__RDIV__0,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1z )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ V0y @ V1z ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__RDIV,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2z )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ V0x @ V2z ) @ ( c_2Ereal_2E_2F @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__lt @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__FRACTION__0,axiom,
! [V0n: tyop_2Enum_2Enum,V1d: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V0n = c_2Enum_2E0 ) )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ V1d @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1d ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__MULTIPLE,axiom,
! [V0n: tyop_2Enum_2Enum,V1d: tyop_2Erealax_2Ereal] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n )
=> ( ( c_2Erealax_2Ereal__lt @ V1d @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ V1d ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1d ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__FRACTION,axiom,
! [V0n: tyop_2Enum_2Enum,V1d: tyop_2Erealax_2Ereal] :
( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ V1d @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) @ V1d )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1d ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__HALF1,axiom,
! [V0d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ V0d @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0d ) ) ).
thf(thm_2Ereal_2EREAL__LT__HALF2,axiom,
! [V0d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ V0d @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0d )
= ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0d ) ) ).
thf(thm_2Ereal_2EREAL__DOUBLE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V0x @ V0x )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__DIV__LMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V1y
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Ereal__mul @ V1y @ ( c_2Ereal_2E_2F @ V0x @ V1y ) )
= V0x ) ) ).
thf(thm_2Ereal_2EREAL__DIV__RMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V1y
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ V0x @ V1y ) @ V1y )
= V0x ) ) ).
thf(thm_2Ereal_2EREAL__HALF__DOUBLE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__DOWN,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ? [V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y )
& ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__DOWN2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ? [V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2z )
& ( c_2Erealax_2Ereal__lt @ V2z @ V0x )
& ( c_2Erealax_2Ereal__lt @ V2z @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) @ V0x )
= ( c_2Erealax_2Ereal__neg @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__LT__ADD__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ V2z )
= ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__sub @ V2z @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__SUB__RADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) @ V2z )
= ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Erealax_2Ereal__add @ V2z @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__SUB__LADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__sub @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V0x @ V2z ) @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__LE__SUB__LADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__sub @ V1y @ V2z ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ V0x @ V2z ) @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__LE__SUB__RADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) @ V2z )
= ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Erealax_2Ereal__add @ V2z @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ V0x ) @ ( c_2Erealax_2Ereal__neg @ V1y ) )
= ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LE__NEG,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__ADD2__SUB2,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__add @ V0a @ V1b ) @ ( c_2Erealax_2Ereal__add @ V2c @ V3d ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__sub @ V0a @ V2c ) @ ( c_2Ereal_2Ereal__sub @ V1b @ V3d ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB__LZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
= ( c_2Erealax_2Ereal__neg @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__SUB__RZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__LET__ADD2,axiom,
! [V0w: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0w @ V1x )
& ( c_2Erealax_2Ereal__lt @ V2y @ V3z ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V0w @ V2y ) @ ( c_2Erealax_2Ereal__add @ V1x @ V3z ) ) ) ).
thf(thm_2Ereal_2EREAL__LTE__ADD2,axiom,
! [V0w: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V0w @ V1x )
& ( c_2Ereal_2Ereal__lte @ V2y @ V3z ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V0w @ V2y ) @ ( c_2Erealax_2Ereal__add @ V1x @ V3z ) ) ) ).
thf(thm_2Ereal_2EREAL__LET__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LTE__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__MUL2,axiom,
! [V0x1: tyop_2Erealax_2Ereal,V1x2: tyop_2Erealax_2Ereal,V2y1: tyop_2Erealax_2Ereal,V3y2: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x1 )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2y1 )
& ( c_2Erealax_2Ereal__lt @ V0x1 @ V1x2 )
& ( c_2Erealax_2Ereal__lt @ V2y1 @ V3y2 ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x1 @ V2y1 ) @ ( c_2Erealax_2Ereal__mul @ V1x2 @ V3y2 ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__INV,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ V1y ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Einv @ V1y ) @ ( c_2Erealax_2Einv @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB__LNEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__neg @ V0x ) @ V1y )
= ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB__RNEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ V0x @ ( c_2Erealax_2Ereal__neg @ V1y ) )
= ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__SUB__NEG2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__neg @ V0x ) @ ( c_2Erealax_2Ereal__neg @ V1y ) )
= ( c_2Ereal_2Ereal__sub @ V1y @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__SUB__TRIANGLE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__sub @ V0a @ V1b ) @ ( c_2Ereal_2Ereal__sub @ V1b @ V2c ) )
= ( c_2Ereal_2Ereal__sub @ V0a @ V2c ) ) ).
thf(thm_2Ereal_2EREAL__EQ__SUB__LADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( V0x
= ( c_2Ereal_2Ereal__sub @ V1y @ V2z ) )
<=> ( ( c_2Erealax_2Ereal__add @ V0x @ V2z )
= V1y ) ) ).
thf(thm_2Ereal_2EREAL__EQ__SUB__RADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__sub @ V0x @ V1y )
= V2z )
<=> ( V0x
= ( c_2Erealax_2Ereal__add @ V2z @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__INV__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V1y
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Erealax_2Einv @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0x ) @ ( c_2Erealax_2Einv @ V1y ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__LMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) )
= ( c_2Ereal_2Ereal__lte @ V1y @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__RMUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2z )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) )
= ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB__INV2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V1y
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Einv @ V0x ) @ ( c_2Erealax_2Einv @ V1y ) )
= ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__sub @ V1y @ V0x ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB__SUB2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ V0x @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) )
= V1y ) ).
thf(thm_2Ereal_2EREAL__ADD__SUB2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ V0x @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__neg @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__MEAN,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
=> ? [V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ V2z )
& ( c_2Erealax_2Ereal__lt @ V2z @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__EQ__LMUL2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( V1y = V2z )
<=> ( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__MUL2,axiom,
! [V0x1: tyop_2Erealax_2Ereal,V1x2: tyop_2Erealax_2Ereal,V2y1: tyop_2Erealax_2Ereal,V3y2: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x1 )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2y1 )
& ( c_2Ereal_2Ereal__lte @ V0x1 @ V1x2 )
& ( c_2Ereal_2Ereal__lte @ V2y1 @ V3y2 ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V0x1 @ V2y1 ) @ ( c_2Erealax_2Ereal__mul @ V1x2 @ V3y2 ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__LDIV,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V1y @ ( c_2Erealax_2Ereal__mul @ V2z @ V0x ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ V1y @ V0x ) @ V2z ) ) ).
thf(thm_2Ereal_2EREAL__LE__RDIV,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V1y @ V0x ) @ V2z ) )
=> ( c_2Ereal_2Ereal__lte @ V1y @ ( c_2Ereal_2E_2F @ V2z @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__DIV,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__DIV,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__1,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ V1y ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ V0x @ V1y ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__LMUL__IMP,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V1y @ V2z ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__RMUL__IMP,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V1y @ V2z ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V1y @ V0x ) @ ( c_2Erealax_2Ereal__mul @ V2z @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__EQ__IMP__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( V0x = V1y )
=> ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__INV__LT1,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Erealax_2Einv @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__POS__NZ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal_2EREAL__EQ__RMUL__IMP,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V2z
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( ( c_2Erealax_2Ereal__mul @ V0x @ V2z )
= ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) ) )
=> ( V0x = V1y ) ) ).
thf(thm_2Ereal_2EREAL__EQ__LMUL__IMP,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) ) )
=> ( V1y = V2z ) ) ).
thf(thm_2Ereal_2EREAL__FACT__NZ,axiom,
! [V0n: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V0n ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EREAL__DIFFSQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ V0x @ V0x ) @ ( c_2Erealax_2Ereal__mul @ V1y @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__POASQ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V0x ) )
<=> ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal_2EREAL__SUMSQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0x @ V0x ) @ ( c_2Erealax_2Ereal__mul @ V1y @ V1y ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( V1y
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal_2EREAL__EQ__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__neg @ V0x )
= ( c_2Erealax_2Ereal__neg @ V1y ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ereal_2EREAL__DIV__MUL2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1z: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V1z
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ! [V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2E_2F @ V2y @ V1z )
= ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__mul @ V0x @ V2y ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V1z ) ) ) ) ).
thf(thm_2Ereal_2EREAL__MIDDLE1,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0a @ V1b )
=> ( c_2Ereal_2Ereal__lte @ V0a @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__add @ V0a @ V1b ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__MIDDLE2,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0a @ V1b )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__add @ V0a @ V1b ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ V1b ) ) ).
thf(thm_2Ereal_2EABS__ZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Eabs @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EABS__0,axiom,
( ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EABS__1,axiom,
( ( c_2Ereal_2Eabs @ ( 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_2Ereal_2EABS__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__neg @ V0x ) )
= ( c_2Ereal_2Eabs @ V0x ) ) ).
thf(thm_2Ereal_2EABS__TRIANGLE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) ) ).
thf(thm_2Ereal_2EABS__TRIANGLE__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) ) ).
thf(thm_2Ereal_2EABS__TRIANGLE__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Eabs @ V1y ) @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) ) ) ).
thf(thm_2Ereal_2EABS__TRIANGLE__LT,axiom,
! [V0e: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Eabs @ V1x ) @ ( c_2Ereal_2Eabs @ V2y ) ) @ V0e )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ V1x @ V2y ) ) @ V0e ) ) ).
thf(thm_2Ereal_2EABS__POS,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Eabs @ V0x ) ) ).
thf(thm_2Ereal_2EABS__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) ) ).
thf(thm_2Ereal_2EABS__LT__MUL2,axiom,
! [V0w: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V0w ) @ V2y )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V1x ) @ V3z ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__mul @ V0w @ V1x ) ) @ ( c_2Erealax_2Ereal__mul @ V2y @ V3z ) ) ) ).
thf(thm_2Ereal_2EABS__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) )
= ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1y @ V0x ) ) ) ).
thf(thm_2Ereal_2EABS__NZ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
<=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Eabs @ V0x ) ) ) ).
thf(thm_2Ereal_2EABS__INV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal_2Eabs @ ( c_2Erealax_2Einv @ V0x ) )
= ( c_2Erealax_2Einv @ ( c_2Ereal_2Eabs @ V0x ) ) ) ) ).
thf(thm_2Ereal_2EABS__ABS,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( c_2Ereal_2Eabs @ V0x ) )
= ( c_2Ereal_2Eabs @ V0x ) ) ).
thf(thm_2Ereal_2EABS__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Eabs @ V0x ) ) ).
thf(thm_2Ereal_2EABS__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Eabs @ V0x )
= V0x )
<=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Ereal_2EABS__N,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__of__num @ V0n ) )
= ( c_2Ereal_2Ereal__of__num @ V0n ) ) ).
thf(thm_2Ereal_2EABS__BETWEEN,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2d: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2d )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__sub @ V0x @ V2d ) @ V1y )
& ( c_2Erealax_2Ereal__lt @ V1y @ ( c_2Erealax_2Ereal__add @ V0x @ V2d ) ) )
<=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1y @ V0x ) ) @ V2d ) ) ).
thf(thm_2Ereal_2EABS__BOUND,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) @ V2d )
=> ( c_2Erealax_2Ereal__lt @ V1y @ ( c_2Erealax_2Ereal__add @ V0x @ V2d ) ) ) ).
thf(thm_2Ereal_2EABS__STILLNZ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) @ ( c_2Ereal_2Eabs @ V1y ) )
=> ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal_2EABS__CASES,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Eabs @ V0x ) ) ) ).
thf(thm_2Ereal_2EABS__BETWEEN1,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V0x @ V2z )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1y @ V0x ) ) @ ( c_2Ereal_2Ereal__sub @ V2z @ V0x ) ) )
=> ( c_2Erealax_2Ereal__lt @ V1y @ V2z ) ) ).
thf(thm_2Ereal_2EABS__SIGN,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) @ V1y )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Ereal_2EABS__SIGN2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) @ ( c_2Erealax_2Ereal__neg @ V1y ) )
=> ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EABS__DIV,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0y
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( c_2Ereal_2E_2F @ V1x @ V0y ) )
= ( c_2Ereal_2E_2F @ ( c_2Ereal_2Eabs @ V1x ) @ ( c_2Ereal_2Eabs @ V0y ) ) ) ) ).
thf(thm_2Ereal_2EABS__CIRCLE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V2h ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Eabs @ V1y ) @ ( c_2Ereal_2Eabs @ V0x ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ V0x @ V2h ) ) @ ( c_2Ereal_2Eabs @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB__ABS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EABS__SUB__ABS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) ) @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EABS__BETWEEN2,axiom,
! [V0x0: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y0: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V0x0 @ V2y0 )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1x @ V0x0 ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__sub @ V2y0 @ V0x0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V3y @ V2y0 ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__sub @ V2y0 @ V0x0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ V1x @ V3y ) ) ).
thf(thm_2Ereal_2EABS__BOUNDS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1k: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V0x ) @ V1k )
<=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ V1k ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ V1k ) ) ) ).
thf(thm_2Ereal_2EPOW__0,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EPOW__NZ,axiom,
! [V0c: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( (~)
@ ( V0c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( (~)
@ ( ( c_2Ereal_2Epow @ V0c @ V1n )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal_2EPOW__INV,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Einv @ ( c_2Ereal_2Epow @ V0c @ V1n ) )
= ( c_2Ereal_2Epow @ ( c_2Erealax_2Einv @ V0c ) @ V1n ) ) ) ).
thf(thm_2Ereal_2EPOW__ABS,axiom,
! [V0c: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ ( c_2Ereal_2Eabs @ V0c ) @ V1n )
= ( c_2Ereal_2Eabs @ ( c_2Ereal_2Epow @ V0c @ V1n ) ) ) ).
thf(thm_2Ereal_2EPOW__PLUS1,axiom,
! [V0e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0e )
=> ! [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ V0e ) ) @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0e ) @ V1n ) ) ) ).
thf(thm_2Ereal_2EPOW__ADD,axiom,
! [V0c: tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ V0c @ ( c_2Earithmetic_2E_2B @ V1m @ V2n ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ V0c @ V1m ) @ ( c_2Ereal_2Epow @ V0c @ V2n ) ) ) ).
thf(thm_2Ereal_2EPOW__1,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= V0x ) ).
thf(thm_2Ereal_2EPOW__2,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Erealax_2Ereal__mul @ V0x @ V0x ) ) ).
thf(thm_2Ereal_2EPOW__ONE,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0n )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ereal_2EPOW__POS,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ! [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Epow @ V0x @ V1n ) ) ) ).
thf(thm_2Ereal_2EPOW__LE,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
& ( c_2Ereal_2Ereal__lte @ V1x @ V2y ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Epow @ V1x @ V0n ) @ ( c_2Ereal_2Epow @ V2y @ V0n ) ) ) ).
thf(thm_2Ereal_2EPOW__M1,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Eabs @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0n ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ereal_2EPOW__MUL,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__mul @ V1x @ V2y ) @ V0n )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ V1x @ V0n ) @ ( c_2Ereal_2Epow @ V2y @ V0n ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__POW2,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ereal_2EABS__POW2,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ereal_2EREAL__POW2__ABS,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epow @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LE1__POW2,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LT1__POW2,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal_2EPOW__POS__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Epow @ V0x @ ( c_2Enum_2ESUC @ V1n ) ) ) ) ).
thf(thm_2Ereal_2EPOW__2__LE1,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0n ) ) ).
thf(thm_2Ereal_2EPOW__2__LT,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0n ) ) ).
thf(thm_2Ereal_2EPOW__MINUS1,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0n ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ereal_2EPOW__LT,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
& ( c_2Erealax_2Ereal__lt @ V1x @ V2y ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Epow @ V1x @ ( c_2Enum_2ESUC @ V0n ) ) @ ( c_2Ereal_2Epow @ V2y @ ( c_2Enum_2ESUC @ V0n ) ) ) ) ).
thf(thm_2Ereal_2EREAL__POW__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Epow @ V0x @ V1n ) ) ) ).
thf(thm_2Ereal_2EPOW__EQ,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2y )
& ( ( c_2Ereal_2Epow @ V1x @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Ereal_2Epow @ V2y @ ( c_2Enum_2ESUC @ V0n ) ) ) )
=> ( V1x = V2y ) ) ).
thf(thm_2Ereal_2EPOW__ZERO,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Epow @ V1x @ V0n )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
=> ( V1x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EPOW__ZERO__EQ,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Epow @ V1x @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( V1x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EREAL__POW__LT2,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( (~) @ ( V0n = c_2Enum_2E0 ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
& ( c_2Erealax_2Ereal__lt @ V1x @ V2y ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Epow @ V1x @ V0n ) @ ( c_2Ereal_2Epow @ V2y @ V0n ) ) ) ).
thf(thm_2Ereal_2EREAL__POW__MONO__LT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2x )
& ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Epow @ V2x @ V0m ) @ ( c_2Ereal_2Epow @ V2x @ V1n ) ) ) ).
thf(thm_2Ereal_2EREAL__POW__POW,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ ( c_2Ereal_2Epow @ V0x @ V1m ) @ V2n )
= ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2E_2A @ V1m @ V2n ) ) ) ).
thf(thm_2Ereal_2EREAL__SUP__SOMEPOS,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ? [V1x: tyop_2Erealax_2Ereal] :
( ( V0P @ V1x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x ) )
& ? [V2z: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0P @ V3x )
=> ( c_2Erealax_2Ereal__lt @ V3x @ V2z ) ) )
=> ? [V4s: tyop_2Erealax_2Ereal] :
! [V5y: tyop_2Erealax_2Ereal] :
( ? [V6x: tyop_2Erealax_2Ereal] :
( ( V0P @ V6x )
& ( c_2Erealax_2Ereal__lt @ V5y @ V6x ) )
<=> ( c_2Erealax_2Ereal__lt @ V5y @ V4s ) ) ) ).
thf(thm_2Ereal_2ESUP__LEMMA1,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1s: tyop_2Erealax_2Ereal,V2d: tyop_2Erealax_2Ereal] :
( ! [V3y: tyop_2Erealax_2Ereal] :
( ? [V4x: tyop_2Erealax_2Ereal] :
( ( ^ [V5x: tyop_2Erealax_2Ereal] : ( V0P @ ( c_2Erealax_2Ereal__add @ V5x @ V2d ) )
@ V4x )
& ( c_2Erealax_2Ereal__lt @ V3y @ V4x ) )
<=> ( c_2Erealax_2Ereal__lt @ V3y @ V1s ) )
=> ! [V6y: tyop_2Erealax_2Ereal] :
( ? [V7x: tyop_2Erealax_2Ereal] :
( ( V0P @ V7x )
& ( c_2Erealax_2Ereal__lt @ V6y @ V7x ) )
<=> ( c_2Erealax_2Ereal__lt @ V6y @ ( c_2Erealax_2Ereal__add @ V1s @ V2d ) ) ) ) ).
thf(thm_2Ereal_2ESUP__LEMMA2,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ? [V1x: tyop_2Erealax_2Ereal] : ( V0P @ V1x )
=> ? [V2d: tyop_2Erealax_2Ereal,V3x: tyop_2Erealax_2Ereal] :
( ( ^ [V4x: tyop_2Erealax_2Ereal] : ( V0P @ ( c_2Erealax_2Ereal__add @ V4x @ V2d ) )
@ V3x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3x ) ) ) ).
thf(thm_2Ereal_2ESUP__LEMMA3,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1d: tyop_2Erealax_2Ereal] :
( ? [V2z: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0P @ V3x )
=> ( c_2Erealax_2Ereal__lt @ V3x @ V2z ) )
=> ? [V4z: tyop_2Erealax_2Ereal] :
! [V5x: tyop_2Erealax_2Ereal] :
( ( ^ [V6x: tyop_2Erealax_2Ereal] : ( V0P @ ( c_2Erealax_2Ereal__add @ V6x @ V1d ) )
@ V5x )
=> ( c_2Erealax_2Ereal__lt @ V5x @ V4z ) ) ) ).
thf(thm_2Ereal_2EREAL__SUP__EXISTS,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ? [V1x: tyop_2Erealax_2Ereal] : ( V0P @ V1x )
& ? [V2z: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0P @ V3x )
=> ( c_2Erealax_2Ereal__lt @ V3x @ V2z ) ) )
=> ? [V4s: tyop_2Erealax_2Ereal] :
! [V5y: tyop_2Erealax_2Ereal] :
( ? [V6x: tyop_2Erealax_2Ereal] :
( ( V0P @ V6x )
& ( c_2Erealax_2Ereal__lt @ V5y @ V6x ) )
<=> ( c_2Erealax_2Ereal__lt @ V5y @ V4s ) ) ) ).
thf(thm_2Ereal_2EREAL__SUP,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ? [V1x: tyop_2Erealax_2Ereal] : ( V0P @ V1x )
& ? [V2z: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0P @ V3x )
=> ( c_2Erealax_2Ereal__lt @ V3x @ V2z ) ) )
=> ! [V4y: tyop_2Erealax_2Ereal] :
( ? [V5x: tyop_2Erealax_2Ereal] :
( ( V0P @ V5x )
& ( c_2Erealax_2Ereal__lt @ V4y @ V5x ) )
<=> ( c_2Erealax_2Ereal__lt @ V4y @ ( c_2Ereal_2Esup @ V0P ) ) ) ) ).
thf(thm_2Ereal_2EREAL__SUP__UBOUND,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ? [V1x: tyop_2Erealax_2Ereal] : ( V0P @ V1x )
& ? [V2z: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0P @ V3x )
=> ( c_2Erealax_2Ereal__lt @ V3x @ V2z ) ) )
=> ! [V4y: tyop_2Erealax_2Ereal] :
( ( V0P @ V4y )
=> ( c_2Ereal_2Ereal__lte @ V4y @ ( c_2Ereal_2Esup @ V0P ) ) ) ) ).
thf(thm_2Ereal_2ESETOK__LE__LT,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ? [V1x: tyop_2Erealax_2Ereal] : ( V0P @ V1x )
& ? [V2z: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0P @ V3x )
=> ( c_2Ereal_2Ereal__lte @ V3x @ V2z ) ) )
<=> ( ? [V4x: tyop_2Erealax_2Ereal] : ( V0P @ V4x )
& ? [V5z: tyop_2Erealax_2Ereal] :
! [V6x: tyop_2Erealax_2Ereal] :
( ( V0P @ V6x )
=> ( c_2Erealax_2Ereal__lt @ V6x @ V5z ) ) ) ) ).
thf(thm_2Ereal_2EREAL__SUP__LE,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ? [V1x: tyop_2Erealax_2Ereal] : ( V0P @ V1x )
& ? [V2z: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0P @ V3x )
=> ( c_2Ereal_2Ereal__lte @ V3x @ V2z ) ) )
=> ! [V4y: tyop_2Erealax_2Ereal] :
( ? [V5x: tyop_2Erealax_2Ereal] :
( ( V0P @ V5x )
& ( c_2Erealax_2Ereal__lt @ V4y @ V5x ) )
<=> ( c_2Erealax_2Ereal__lt @ V4y @ ( c_2Ereal_2Esup @ V0P ) ) ) ) ).
thf(thm_2Ereal_2EREAL__SUP__UBOUND__LE,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ? [V1x: tyop_2Erealax_2Ereal] : ( V0P @ V1x )
& ? [V2z: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0P @ V3x )
=> ( c_2Ereal_2Ereal__lte @ V3x @ V2z ) ) )
=> ! [V4y: tyop_2Erealax_2Ereal] :
( ( V0P @ V4y )
=> ( c_2Ereal_2Ereal__lte @ V4y @ ( c_2Ereal_2Esup @ V0P ) ) ) ) ).
thf(thm_2Ereal_2EREAL__ARCH,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ! [V1y: tyop_2Erealax_2Ereal] :
? [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ V1y @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2n ) @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__ARCH__LEAST,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0y )
=> ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
=> ? [V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2n ) @ V0y ) @ V1x )
& ( c_2Erealax_2Ereal__lt @ V1x @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Enum_2ESUC @ V2n ) ) @ V0y ) ) ) ) ) ).
thf(thm_2Ereal_2Esum__ind,axiom,
! [V0P: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o] :
( ( ! [V1n: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1n @ c_2Enum_2E0 ) @ V2f )
& ! [V3n: tyop_2Enum_2Enum,V4m: tyop_2Enum_2Enum,V5f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V3n @ V4m ) @ V5f )
=> ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V3n @ ( c_2Enum_2ESUC @ V4m ) ) @ V5f ) ) )
=> ! [V6v: tyop_2Enum_2Enum,V7v1: tyop_2Enum_2Enum,V8v2: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V6v @ V7v1 ) @ V8v2 ) ) ).
thf(thm_2Ereal_2Esum,axiom,
( ! [V0n: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0n @ c_2Enum_2E0 ) @ V1f )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ! [V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum,V4f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2n @ ( c_2Enum_2ESUC @ V3m ) ) @ V4f )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2n @ V3m ) @ V4f ) @ ( V4f @ ( c_2Earithmetic_2E_2B @ V2n @ V3m ) ) ) ) ) ).
thf(thm_2Ereal_2Esum__compute,axiom,
( ! [V0n: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0n @ c_2Enum_2E0 ) @ V1f )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ! [V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum,V4f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3m ) ) ) @ V4f )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2n @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3m ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V4f ) @ ( V4f @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3m ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ! [V5n: tyop_2Enum_2Enum,V6m: tyop_2Enum_2Enum,V7f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V5n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V6m ) ) ) @ V7f )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V5n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V6m ) ) ) @ V7f ) @ ( V7f @ ( c_2Earithmetic_2E_2B @ V5n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V6m ) ) ) ) ) ) ) ).
thf(thm_2Ereal_2ESUM__TWO,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1n ) @ V0f ) @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1n @ V2p ) @ V0f ) )
= ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) ) @ V0f ) ) ).
thf(thm_2Ereal_2ESUM__DIFF,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V2n ) @ V0f )
= ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2B @ V1m @ V2n ) ) @ V0f ) @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1m ) @ V0f ) ) ) ).
thf(thm_2Ereal_2EABS__SUM,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V2n ) @ V0f ) )
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V2n )
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0f @ V3n ) ) ) ) ).
thf(thm_2Ereal_2ESUM__LE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ! [V4r: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V4r )
& ( c_2Eprim__rec_2E_3C @ V4r @ ( c_2Earithmetic_2E_2B @ V3n @ V2m ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V4r ) @ ( V1g @ V4r ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V0f ) @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Ereal_2ESUM__EQ,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ! [V4r: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V4r )
& ( c_2Eprim__rec_2E_3C @ V4r @ ( c_2Earithmetic_2E_2B @ V3n @ V2m ) ) )
=> ( ( V0f @ V4r )
= ( V1g @ V4r ) ) )
=> ( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V0f )
= ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Ereal_2ESUM__POS,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ! [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V1n ) )
=> ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V0f ) ) ) ).
thf(thm_2Ereal_2ESUM__POS__GEN,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum] :
( ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V2n ) ) )
=> ! [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V3n ) @ V0f ) ) ) ).
thf(thm_2Ereal_2ESUM__ABS,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Eabs
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V2n )
@ ^ [V3m: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0f @ V3m ) ) ) )
= ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V2n )
@ ^ [V4m: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0f @ V4m ) ) ) ) ).
thf(thm_2Ereal_2ESUM__ABS__LE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V2n ) @ V0f ) )
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V2n )
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0f @ V3n ) ) ) ) ).
thf(thm_2Ereal_2ESUM__ZERO,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1N: tyop_2Enum_2Enum] :
( ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V2n @ V1N )
=> ( ( V0f @ V2n )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V3m @ V1N )
=> ( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V3m @ V4n ) @ V0f )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ereal_2ESUM__ADD,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n )
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add @ ( V0f @ V4n ) @ ( V1g @ V4n ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V0f ) @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Ereal_2ESUM__CMUL,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n )
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ V1c @ ( V0f @ V4n ) ) )
= ( c_2Erealax_2Ereal__mul @ V1c @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V0f ) ) ) ).
thf(thm_2Ereal_2ESUM__NEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum,V2d: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1n @ V2d )
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( V0f @ V3n ) ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1n @ V2d ) @ V0f ) ) ) ).
thf(thm_2Ereal_2ESUM__SUB,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n )
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0f @ V4n ) @ ( V1g @ V4n ) ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V0f ) @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Ereal_2ESUM__SUBST,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ! [V4p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V4p )
& ( c_2Eprim__rec_2E_3C @ V4p @ ( c_2Earithmetic_2E_2B @ V2m @ V3n ) ) )
=> ( ( V0f @ V4p )
= ( V1g @ V4p ) ) )
=> ( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V0f )
= ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Ereal_2ESUM__NSUB,axiom,
! [V0n: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0n ) @ V1f ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ V2c ) )
= ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0n )
@ ^ [V3p: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V1f @ V3p ) @ V2c ) ) ) ).
thf(thm_2Ereal_2ESUM__BOUND,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k: tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ! [V4p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V4p )
& ( c_2Eprim__rec_2E_3C @ V4p @ ( c_2Earithmetic_2E_2B @ V2m @ V3n ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V4p ) @ V1k ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2m @ V3n ) @ V0f ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V3n ) @ V1k ) ) ) ).
thf(thm_2Ereal_2ESUM__GROUP,axiom,
! [V0n: tyop_2Enum_2Enum,V1k: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0n )
@ ^ [V3m: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_2A @ V3m @ V1k ) @ V1k ) @ V2f ) )
= ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2A @ V0n @ V1k ) ) @ V2f ) ) ).
thf(thm_2Ereal_2ESUM__1,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0f )
= ( V0f @ V1n ) ) ).
thf(thm_2Ereal_2ESUM__2,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0f )
= ( c_2Erealax_2Ereal__add @ ( V0f @ V1n ) @ ( V0f @ ( c_2Earithmetic_2E_2B @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ereal_2ESUM__OFFSET,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum,V2k: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1n )
@ ^ [V3m: tyop_2Enum_2Enum] : ( V0f @ ( c_2Earithmetic_2E_2B @ V3m @ V2k ) ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2B @ V1n @ V2k ) ) @ V0f ) @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V2k ) @ V0f ) ) ) ).
thf(thm_2Ereal_2ESUM__REINDEX,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2k: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_2B @ V1m @ V2k ) @ V3n ) @ V0f )
= ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1m @ V3n )
@ ^ [V4r: tyop_2Enum_2Enum] : ( V0f @ ( c_2Earithmetic_2E_2B @ V4r @ V2k ) ) ) ) ).
thf(thm_2Ereal_2ESUM__0,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0m @ V1n )
@ ^ [V2r: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2ESUM__PERMUTE__0,axiom,
! [V0n: tyop_2Enum_2Enum,V1p: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V2y: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2y @ V0n )
=> ( c_2Ebool_2E_3F_21 @ tyop_2Enum_2Enum
@ ^ [V3x: tyop_2Enum_2Enum] : ( c_2Ebool_2E_2F_5C @ ( c_2Eprim__rec_2E_3C @ V3x @ V0n ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ ( V1p @ V3x ) @ V2y ) ) ) )
=> ! [V4f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0n )
@ ^ [V5n: tyop_2Enum_2Enum] : ( V4f @ ( V1p @ V5n ) ) )
= ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0n ) @ V4f ) ) ) ).
thf(thm_2Ereal_2ESUM__CANCEL,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum,V2d: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1n @ V2d )
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0f @ ( c_2Enum_2ESUC @ V3n ) ) @ ( V0f @ V3n ) ) )
= ( c_2Ereal_2Ereal__sub @ ( V0f @ ( c_2Earithmetic_2E_2B @ V1n @ V2d ) ) @ ( V0f @ 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__POW__INV,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ ( c_2Erealax_2Einv @ V0x ) @ V1n )
= ( c_2Erealax_2Einv @ ( c_2Ereal_2Epow @ V0x @ V1n ) ) ) ).
thf(thm_2Ereal_2EREAL__POW__DIV,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ ( c_2Ereal_2E_2F @ V0x @ V1y ) @ V2n )
= ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ V0x @ V2n ) @ ( c_2Ereal_2Epow @ V1y @ V2n ) ) ) ).
thf(thm_2Ereal_2EREAL__POW__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2E_2B @ V1m @ V2n ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ V0x @ V1m ) @ ( c_2Ereal_2Epow @ V0x @ V2n ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__RDIV__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2z )
=> ( ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2E_2F @ V1y @ V2z ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__LDIV__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2z )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ V0x @ V2z ) @ V1y )
= ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__RDIV__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2z )
=> ( ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2E_2F @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__LDIV__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2z )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ V0x @ V2z ) @ V1y )
= ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) ) ) ) ).
thf(thm_2Ereal_2EREAL__EQ__RDIV__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2z )
=> ( ( V0x
= ( c_2Ereal_2E_2F @ V1y @ V2z ) )
<=> ( ( c_2Erealax_2Ereal__mul @ V0x @ V2z )
= V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__EQ__LDIV__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2z )
=> ( ( ( c_2Ereal_2E_2F @ V0x @ V2z )
= V1y )
<=> ( V0x
= ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) ) ) ) ).
thf(thm_2Ereal_2EREAL__OF__NUM__POW,axiom,
! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ V0x ) @ V1n )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EEXP @ V0x @ V1n ) ) ) ).
thf(thm_2Ereal_2EREAL__ADD__LDISTRIB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Ereal__add @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) ) ) ).
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_2EREAL__OF__NUM__SUC,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Enum_2ESUC @ V0n ) ) ) ).
thf(thm_2Ereal_2EREAL__OF__NUM__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Ereal_2Ereal__of__num @ V0m )
= ( c_2Ereal_2Ereal__of__num @ V1n ) )
<=> ( V0m = V1n ) ) ).
thf(thm_2Ereal_2EREAL__EQ__MUL__LCANCEL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) )
<=> ( ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( V1y = V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__ABS__0,axiom,
( ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2EREAL__ABS__TRIANGLE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__ABS__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Eabs @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__ABS__POS,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Eabs @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__LE__EPSILON,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Erealax_2Ereal__add @ V1y @ V2e ) ) )
=> ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__BIGNUM,axiom,
! [V0r: tyop_2Erealax_2Ereal] :
? [V1n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ V0r @ ( c_2Ereal_2Ereal__of__num @ V1n ) ) ).
thf(thm_2Ereal_2EREAL__INV__LT__ANTIMONO,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1y ) )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Einv @ V0x ) @ ( c_2Erealax_2Einv @ V1y ) )
= ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__INV__INJ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Einv @ V0x )
= ( c_2Erealax_2Einv @ V1y ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ereal_2EREAL__DIV__RMUL__CANCEL,axiom,
! [V0c: tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__mul @ V1a @ V0c ) @ ( c_2Erealax_2Ereal__mul @ V2b @ V0c ) )
= ( c_2Ereal_2E_2F @ V1a @ V2b ) ) ) ).
thf(thm_2Ereal_2EREAL__DIV__LMUL__CANCEL,axiom,
! [V0c: tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0c
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__mul @ V0c @ V1a ) @ ( c_2Erealax_2Ereal__mul @ V0c @ V2b ) )
= ( c_2Ereal_2E_2F @ V1a @ V2b ) ) ) ).
thf(thm_2Ereal_2EREAL__DIV__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2E_2F @ V1y @ V0x ) @ ( c_2Ereal_2E_2F @ V2z @ V0x ) )
= ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__add @ V1y @ V2z ) @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__ADD__RAT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V1b
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V3d
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2E_2F @ V0a @ V1b ) @ ( c_2Ereal_2E_2F @ V2c @ V3d ) )
= ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0a @ V3d ) @ ( c_2Erealax_2Ereal__mul @ V1b @ V2c ) ) @ ( c_2Erealax_2Ereal__mul @ V1b @ V3d ) ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB__RAT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V1b
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V3d
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2E_2F @ V0a @ V1b ) @ ( c_2Ereal_2E_2F @ V2c @ V3d ) )
= ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ V0a @ V3d ) @ ( c_2Erealax_2Ereal__mul @ V1b @ V2c ) ) @ ( c_2Erealax_2Ereal__mul @ V1b @ V3d ) ) ) ) ).
thf(thm_2Ereal_2EREAL__SUB,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ V0m ) @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2D @ V1n @ V0m ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2D @ V0m @ V1n ) ) ) ) ).
thf(thm_2Ereal_2EREAL__POS__POS,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Epos @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__POS__ID,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( ( c_2Ereal_2Epos @ V0x )
= V0x ) ) ).
thf(thm_2Ereal_2EREAL__POS__INFLATE,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Epos @ V0x ) ) ).
thf(thm_2Ereal_2EREAL__POS__MONO,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Epos @ V0x ) @ ( c_2Ereal_2Epos @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__POS__EQ__ZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Epos @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EREAL__POS__LE__ZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Epos @ V0x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2EREAL__MIN__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Emin @ V0x @ V0x )
= V0x ) ).
thf(thm_2Ereal_2EREAL__LE__MIN,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0z @ ( c_2Ereal_2Emin @ V1x @ V2y ) )
<=> ( ( c_2Ereal_2Ereal__lte @ V0z @ V1x )
& ( c_2Ereal_2Ereal__lte @ V0z @ V2y ) ) ) ).
thf(thm_2Ereal_2EREAL__MIN__LE,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Emin @ V1x @ V2y ) @ V0z )
<=> ( ( c_2Ereal_2Ereal__lte @ V1x @ V0z )
| ( c_2Ereal_2Ereal__lte @ V2y @ V0z ) ) ) ).
thf(thm_2Ereal_2EREAL__MIN__LE1,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Emin @ V0x @ V1y ) @ V0x ) ).
thf(thm_2Ereal_2EREAL__MIN__LE2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Emin @ V0x @ V1y ) @ V1y ) ).
thf(thm_2Ereal_2EREAL__LT__MIN,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V2z @ ( c_2Ereal_2Emin @ V0x @ V1y ) )
<=> ( ( c_2Erealax_2Ereal__lt @ V2z @ V0x )
& ( c_2Erealax_2Ereal__lt @ V2z @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__MIN__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Emin @ V0x @ V1y ) @ V2z )
<=> ( ( c_2Erealax_2Ereal__lt @ V0x @ V2z )
| ( c_2Erealax_2Ereal__lt @ V1y @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__MIN__ALT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
=> ( ( c_2Ereal_2Emin @ V0x @ V1y )
= V0x ) )
& ( ( c_2Ereal_2Ereal__lte @ V1y @ V0x )
=> ( ( c_2Ereal_2Emin @ V0x @ V1y )
= V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__MIN__LE__LIN,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0z )
& ( c_2Ereal_2Ereal__lte @ V0z @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Emin @ V1x @ V2y ) @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0z @ V1x ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0z ) @ V2y ) ) ) ) ).
thf(thm_2Ereal_2EREAL__MIN__ADD,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Emin @ ( c_2Erealax_2Ereal__add @ V1x @ V0z ) @ ( c_2Erealax_2Ereal__add @ V2y @ V0z ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Emin @ V1x @ V2y ) @ V0z ) ) ).
thf(thm_2Ereal_2EREAL__MIN__SUB,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Emin @ ( c_2Ereal_2Ereal__sub @ V1x @ V0z ) @ ( c_2Ereal_2Ereal__sub @ V2y @ V0z ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Emin @ V1x @ V2y ) @ V0z ) ) ).
thf(thm_2Ereal_2EREAL__IMP__MIN__LE2,axiom,
! [V0x1: tyop_2Erealax_2Ereal,V1x2: tyop_2Erealax_2Ereal,V2y1: tyop_2Erealax_2Ereal,V3y2: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0x1 @ V2y1 )
& ( c_2Ereal_2Ereal__lte @ V1x2 @ V3y2 ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Emin @ V0x1 @ V1x2 ) @ ( c_2Ereal_2Emin @ V2y1 @ V3y2 ) ) ) ).
thf(thm_2Ereal_2EREAL__MIN__ACI,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Emin @ V0x @ V1y )
= ( c_2Ereal_2Emin @ V1y @ V0x ) )
& ( ( c_2Ereal_2Emin @ ( c_2Ereal_2Emin @ V0x @ V1y ) @ V2z )
= ( c_2Ereal_2Emin @ V0x @ ( c_2Ereal_2Emin @ V1y @ V2z ) ) )
& ( ( c_2Ereal_2Emin @ V0x @ ( c_2Ereal_2Emin @ V1y @ V2z ) )
= ( c_2Ereal_2Emin @ V1y @ ( c_2Ereal_2Emin @ V0x @ V2z ) ) )
& ( ( c_2Ereal_2Emin @ V0x @ V0x )
= V0x )
& ( ( c_2Ereal_2Emin @ V0x @ ( c_2Ereal_2Emin @ V0x @ V1y ) )
= ( c_2Ereal_2Emin @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__MAX__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Emax @ V0x @ V0x )
= V0x ) ).
thf(thm_2Ereal_2EREAL__LE__MAX,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0z @ ( c_2Ereal_2Emax @ V1x @ V2y ) )
<=> ( ( c_2Ereal_2Ereal__lte @ V0z @ V1x )
| ( c_2Ereal_2Ereal__lte @ V0z @ V2y ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__MAX1,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Emax @ V0x @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__LE__MAX2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ V1y @ ( c_2Ereal_2Emax @ V0x @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__MAX__LE,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Emax @ V1x @ V2y ) @ V0z )
<=> ( ( c_2Ereal_2Ereal__lte @ V1x @ V0z )
& ( c_2Ereal_2Ereal__lte @ V2y @ V0z ) ) ) ).
thf(thm_2Ereal_2EREAL__LT__MAX,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V2z @ ( c_2Ereal_2Emax @ V0x @ V1y ) )
<=> ( ( c_2Erealax_2Ereal__lt @ V2z @ V0x )
| ( c_2Erealax_2Ereal__lt @ V2z @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__MAX__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Emax @ V0x @ V1y ) @ V2z )
<=> ( ( c_2Erealax_2Ereal__lt @ V0x @ V2z )
& ( c_2Erealax_2Ereal__lt @ V1y @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__MAX__ALT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0x @ V1y )
=> ( ( c_2Ereal_2Emax @ V0x @ V1y )
= V1y ) )
& ( ( c_2Ereal_2Ereal__lte @ V1y @ V0x )
=> ( ( c_2Ereal_2Emax @ V0x @ V1y )
= V0x ) ) ) ).
thf(thm_2Ereal_2EREAL__MAX__MIN,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Emax @ V0x @ V1y )
= ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Emin @ ( c_2Erealax_2Ereal__neg @ V0x ) @ ( c_2Erealax_2Ereal__neg @ V1y ) ) ) ) ).
thf(thm_2Ereal_2EREAL__MIN__MAX,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Emin @ V0x @ V1y )
= ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Emax @ ( c_2Erealax_2Ereal__neg @ V0x ) @ ( c_2Erealax_2Ereal__neg @ V1y ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LIN__LE__MAX,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0z )
& ( c_2Ereal_2Ereal__lte @ V0z @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0z @ V1x ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0z ) @ V2y ) ) @ ( c_2Ereal_2Emax @ V1x @ V2y ) ) ) ).
thf(thm_2Ereal_2EREAL__MAX__ADD,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Emax @ ( c_2Erealax_2Ereal__add @ V1x @ V0z ) @ ( c_2Erealax_2Ereal__add @ V2y @ V0z ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Emax @ V1x @ V2y ) @ V0z ) ) ).
thf(thm_2Ereal_2EREAL__MAX__SUB,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Emax @ ( c_2Ereal_2Ereal__sub @ V1x @ V0z ) @ ( c_2Ereal_2Ereal__sub @ V2y @ V0z ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Emax @ V1x @ V2y ) @ V0z ) ) ).
thf(thm_2Ereal_2EREAL__IMP__MAX__LE2,axiom,
! [V0x1: tyop_2Erealax_2Ereal,V1x2: tyop_2Erealax_2Ereal,V2y1: tyop_2Erealax_2Ereal,V3y2: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0x1 @ V2y1 )
& ( c_2Ereal_2Ereal__lte @ V1x2 @ V3y2 ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Emax @ V0x1 @ V1x2 ) @ ( c_2Ereal_2Emax @ V2y1 @ V3y2 ) ) ) ).
thf(thm_2Ereal_2EREAL__MAX__ACI,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Emax @ V0x @ V1y )
= ( c_2Ereal_2Emax @ V1y @ V0x ) )
& ( ( c_2Ereal_2Emax @ ( c_2Ereal_2Emax @ V0x @ V1y ) @ V2z )
= ( c_2Ereal_2Emax @ V0x @ ( c_2Ereal_2Emax @ V1y @ V2z ) ) )
& ( ( c_2Ereal_2Emax @ V0x @ ( c_2Ereal_2Emax @ V1y @ V2z ) )
= ( c_2Ereal_2Emax @ V1y @ ( c_2Ereal_2Emax @ V0x @ V2z ) ) )
& ( ( c_2Ereal_2Emax @ V0x @ V0x )
= V0x )
& ( ( c_2Ereal_2Emax @ V0x @ ( c_2Ereal_2Emax @ V0x @ V1y ) )
= ( c_2Ereal_2Emax @ V0x @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__SUP__EXISTS__UNIQUE,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o] :
( ( ? [V1x: tyop_2Erealax_2Ereal] : ( V0p @ V1x )
& ? [V2z: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0p @ V3x )
=> ( c_2Ereal_2Ereal__lte @ V3x @ V2z ) ) )
=> ( c_2Ebool_2E_3F_21 @ tyop_2Erealax_2Ereal
@ ^ [V4s: tyop_2Erealax_2Ereal] :
( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V5y: tyop_2Erealax_2Ereal] :
( c_2Emin_2E_3D @ $o
@ ( c_2Ebool_2E_3F @ tyop_2Erealax_2Ereal
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( V0p @ V6x ) @ ( c_2Erealax_2Ereal__lt @ V5y @ V6x ) ) )
@ ( c_2Erealax_2Ereal__lt @ V5y @ V4s ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__SUP__MAX,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1z: tyop_2Erealax_2Ereal] :
( ( ( V0p @ V1z )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( V0p @ V2x )
=> ( c_2Ereal_2Ereal__lte @ V2x @ V1z ) ) )
=> ( ( c_2Ereal_2Esup @ V0p )
= V1z ) ) ).
thf(thm_2Ereal_2EREAL__IMP__SUP__LE,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ? [V2r: tyop_2Erealax_2Ereal] : ( V0p @ V2r )
& ! [V3r: tyop_2Erealax_2Ereal] :
( ( V0p @ V3r )
=> ( c_2Ereal_2Ereal__lte @ V3r @ V1x ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esup @ V0p ) @ V1x ) ) ).
thf(thm_2Ereal_2EREAL__IMP__LE__SUP,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ? [V2r: tyop_2Erealax_2Ereal] : ( V0p @ V2r )
& ? [V3z: tyop_2Erealax_2Ereal] :
! [V4r: tyop_2Erealax_2Ereal] :
( ( V0p @ V4r )
=> ( c_2Ereal_2Ereal__lte @ V4r @ V3z ) )
& ? [V5r: tyop_2Erealax_2Ereal] :
( ( V0p @ V5r )
& ( c_2Ereal_2Ereal__lte @ V1x @ V5r ) ) )
=> ( c_2Ereal_2Ereal__lte @ V1x @ ( c_2Ereal_2Esup @ V0p ) ) ) ).
thf(thm_2Ereal_2EREAL__INF__MIN,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1z: tyop_2Erealax_2Ereal] :
( ( ( V0p @ V1z )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( V0p @ V2x )
=> ( c_2Ereal_2Ereal__lte @ V1z @ V2x ) ) )
=> ( ( c_2Ereal_2Einf @ V0p )
= V1z ) ) ).
thf(thm_2Ereal_2EREAL__IMP__LE__INF,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1r: tyop_2Erealax_2Ereal] :
( ( ? [V2x: tyop_2Erealax_2Ereal] : ( V0p @ V2x )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( V0p @ V3x )
=> ( c_2Ereal_2Ereal__lte @ V1r @ V3x ) ) )
=> ( c_2Ereal_2Ereal__lte @ V1r @ ( c_2Ereal_2Einf @ V0p ) ) ) ).
thf(thm_2Ereal_2EREAL__IMP__INF__LE,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1r: tyop_2Erealax_2Ereal] :
( ( ? [V2z: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0p @ V3x )
=> ( c_2Ereal_2Ereal__lte @ V2z @ V3x ) )
& ? [V4x: tyop_2Erealax_2Ereal] :
( ( V0p @ V4x )
& ( c_2Ereal_2Ereal__lte @ V4x @ V1r ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Einf @ V0p ) @ V1r ) ) ).
thf(thm_2Ereal_2EREAL__INF__LT,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1z: tyop_2Erealax_2Ereal] :
( ( ? [V2x: tyop_2Erealax_2Ereal] : ( V0p @ V2x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Einf @ V0p ) @ V1z ) )
=> ? [V3x: tyop_2Erealax_2Ereal] :
( ( V0p @ V3x )
& ( c_2Erealax_2Ereal__lt @ V3x @ V1z ) ) ) ).
thf(thm_2Ereal_2EREAL__INF__CLOSE,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1e: tyop_2Erealax_2Ereal] :
( ( ? [V2x: tyop_2Erealax_2Ereal] : ( V0p @ V2x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e ) )
=> ? [V3x: tyop_2Erealax_2Ereal] :
( ( V0p @ V3x )
& ( c_2Erealax_2Ereal__lt @ V3x @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Einf @ V0p ) @ V1e ) ) ) ) ).
thf(thm_2Ereal_2ESUP__EPSILON,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1e: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e )
& ? [V2x: tyop_2Erealax_2Ereal] : ( V0p @ V2x )
& ? [V3z: tyop_2Erealax_2Ereal] :
! [V4x: tyop_2Erealax_2Ereal] :
( ( V0p @ V4x )
=> ( c_2Ereal_2Ereal__lte @ V4x @ V3z ) ) )
=> ? [V5x: tyop_2Erealax_2Ereal] :
( ( V0p @ V5x )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esup @ V0p ) @ ( c_2Erealax_2Ereal__add @ V5x @ V1e ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__SUP,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ? [V2y: tyop_2Erealax_2Ereal] : ( V0p @ V2y )
& ? [V3y: tyop_2Erealax_2Ereal] :
! [V4z: tyop_2Erealax_2Ereal] :
( ( V0p @ V4z )
=> ( c_2Ereal_2Ereal__lte @ V4z @ V3y ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ V1x @ ( c_2Ereal_2Esup @ V0p ) )
<=> ! [V5y: tyop_2Erealax_2Ereal] :
( ! [V6z: tyop_2Erealax_2Ereal] :
( ( V0p @ V6z )
=> ( c_2Ereal_2Ereal__lte @ V6z @ V5y ) )
=> ( c_2Ereal_2Ereal__lte @ V1x @ V5y ) ) ) ) ).
thf(thm_2Ereal_2EREAL__INF__LE,axiom,
! [V0p: tyop_2Erealax_2Ereal > $o,V1x: tyop_2Erealax_2Ereal] :
( ( ? [V2y: tyop_2Erealax_2Ereal] : ( V0p @ V2y )
& ? [V3y: tyop_2Erealax_2Ereal] :
! [V4z: tyop_2Erealax_2Ereal] :
( ( V0p @ V4z )
=> ( c_2Ereal_2Ereal__lte @ V3y @ V4z ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Einf @ V0p ) @ V1x )
<=> ! [V5y: tyop_2Erealax_2Ereal] :
( ! [V6z: tyop_2Erealax_2Ereal] :
( ( V0p @ V6z )
=> ( c_2Ereal_2Ereal__lte @ V5y @ V6z ) )
=> ( c_2Ereal_2Ereal__lte @ V5y @ V1x ) ) ) ) ).
thf(thm_2Ereal_2EREAL__SUP__CONST,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esup
@ ^ [V1r: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V1r @ V0x ) )
= V0x ) ).
thf(thm_2Ereal_2EREAL__MUL__SUB2__CANCEL,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V1x @ V2y ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ V0z @ V1x ) @ V2y ) )
= ( c_2Erealax_2Ereal__mul @ V0z @ V2y ) ) ).
thf(thm_2Ereal_2EREAL__MUL__SUB1__CANCEL,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V2y @ V1x ) @ ( c_2Erealax_2Ereal__mul @ V2y @ ( c_2Ereal_2Ereal__sub @ V0z @ V1x ) ) )
= ( c_2Erealax_2Ereal__mul @ V2y @ V0z ) ) ).
thf(thm_2Ereal_2EREAL__NEG__HALF,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ V0x @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__NEG__THIRD,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ V0x @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__DIV__DENOM__CANCEL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal_2E_2F @ ( c_2Ereal_2E_2F @ V1y @ V0x ) @ ( c_2Ereal_2E_2F @ V2z @ V0x ) )
= ( c_2Ereal_2E_2F @ V1y @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__DIV__DENOM__CANCEL2,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2E_2F @ ( c_2Ereal_2E_2F @ V0y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2E_2F @ V1z @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Ereal_2E_2F @ V0y @ V1z ) ) ).
thf(thm_2Ereal_2EREAL__DIV__DENOM__CANCEL3,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2E_2F @ ( c_2Ereal_2E_2F @ V0y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Ereal_2E_2F @ V1z @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( c_2Ereal_2E_2F @ V0y @ V1z ) ) ).
thf(thm_2Ereal_2EREAL__DIV__INNER__CANCEL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ V1y @ V0x ) @ ( c_2Ereal_2E_2F @ V0x @ V2z ) )
= ( c_2Ereal_2E_2F @ V1y @ V2z ) ) ) ).
thf(thm_2Ereal_2EREAL__DIV__INNER__CANCEL2,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ V0y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V1z ) )
= ( c_2Ereal_2E_2F @ V0y @ V1z ) ) ).
thf(thm_2Ereal_2EREAL__DIV__INNER__CANCEL3,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ V0y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V1z ) )
= ( c_2Ereal_2E_2F @ V0y @ V1z ) ) ).
thf(thm_2Ereal_2EREAL__DIV__OUTER__CANCEL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ V0x @ V1y ) @ ( c_2Ereal_2E_2F @ V2z @ V0x ) )
= ( c_2Ereal_2E_2F @ V2z @ V1y ) ) ) ).
thf(thm_2Ereal_2EREAL__DIV__OUTER__CANCEL2,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0y ) @ ( c_2Ereal_2E_2F @ V1z @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Ereal_2E_2F @ V1z @ V0y ) ) ).
thf(thm_2Ereal_2EREAL__DIV__OUTER__CANCEL3,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0y ) @ ( c_2Ereal_2E_2F @ V1z @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( c_2Ereal_2E_2F @ V1z @ V0y ) ) ).
thf(thm_2Ereal_2EREAL__DIV__REFL2,axiom,
( ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ereal_2EREAL__DIV__REFL3,axiom,
( ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ereal_2EREAL__HALF__BETWEEN,axiom,
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__THIRDS__BETWEEN,axiom,
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ ( 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_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ ( 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_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ ( 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_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ ( 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_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal_2EREAL__LE__SUB__CANCEL2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__sub @ V0x @ V2z ) @ ( c_2Ereal_2Ereal__sub @ V1y @ V2z ) )
= ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ).
thf(thm_2Ereal_2EREAL__ADD__SUB__ALT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ V1y )
= V0x ) ).
thf(thm_2Ereal_2EINFINITE__REAL__UNIV,axiom,
(~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Ereal_2Eadd__rat,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2v: tyop_2Erealax_2Ereal,V3u: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2E_2F @ V1x @ V0y ) @ ( c_2Ereal_2E_2F @ V3u @ V2v ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1x @ V0y ) ) @ ( c_2Ereal_2E_2F @ V3u @ V2v ) ) @ ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V2v @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2E_2F @ V1x @ V0y ) @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V3u @ V2v ) ) ) @ ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0y @ V2v ) @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__add @ V1x @ V3u ) @ V2v ) @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V1x @ V2v ) @ ( c_2Erealax_2Ereal__mul @ V3u @ V0y ) ) @ ( c_2Erealax_2Ereal__mul @ V0y @ V2v ) ) ) ) ) ) ).
thf(thm_2Ereal_2Eadd__ratl,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2E_2F @ V2x @ V1y ) @ V0z )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V2x @ V1y ) ) @ V0z ) @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__add @ V2x @ ( c_2Erealax_2Ereal__mul @ V0z @ V1y ) ) @ V1y ) ) ) ).
thf(thm_2Ereal_2Eadd__ratr,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V2x @ ( c_2Ereal_2E_2F @ V1y @ V0z ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0z @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Erealax_2Ereal__add @ V2x @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1y @ V0z ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V2x @ V0z ) @ V1y ) @ V0z ) ) ) ).
thf(thm_2Ereal_2Eadd__ints,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Ereal_2Ereal__of__num @ V1m ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) )
& ( ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) @ ( c_2Ereal_2Ereal__of__num @ V1m ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2E_3C_3D @ V0n @ V1m ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2D @ V1m @ V0n ) ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2D @ V0n @ V1m ) ) ) ) )
& ( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V1m ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Eprim__rec_2E_3C @ V0n @ V1m ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2D @ V1m @ V0n ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2D @ V0n @ V1m ) ) ) )
& ( ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V1m ) ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) ) ) ) ) ).
thf(thm_2Ereal_2Emult__rat,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2v: tyop_2Erealax_2Ereal,V3u: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ V1x @ V0y ) @ ( c_2Ereal_2E_2F @ V3u @ V2v ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1x @ V0y ) ) @ ( c_2Ereal_2E_2F @ V3u @ V2v ) ) @ ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V2v @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ V1x @ V0y ) @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V3u @ V2v ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__mul @ V1x @ V3u ) @ ( c_2Erealax_2Ereal__mul @ V0y @ V2v ) ) ) ) ) ).
thf(thm_2Ereal_2Emult__ratl,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ V2x @ V1y ) @ V0z )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V2x @ V1y ) ) @ V0z ) @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__mul @ V2x @ V0z ) @ V1y ) ) ) ).
thf(thm_2Ereal_2Emult__ratr,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V2x @ ( c_2Ereal_2E_2F @ V1y @ V0z ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0z @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Erealax_2Ereal__mul @ V2x @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1y @ V0z ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__mul @ V2x @ V1y ) @ V0z ) ) ) ).
thf(thm_2Ereal_2Emult__ints,axiom,
! [V0b: tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum] :
( ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1a ) @ ( c_2Ereal_2Ereal__of__num @ V0b ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2A @ V1a @ V0b ) ) )
& ( ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V1a ) ) @ ( c_2Ereal_2Ereal__of__num @ V0b ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2A @ V1a @ V0b ) ) ) )
& ( ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1a ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V0b ) ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2A @ V1a @ V0b ) ) ) )
& ( ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V1a ) ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V0b ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2A @ V1a @ V0b ) ) ) ) ).
thf(thm_2Ereal_2Epow__rat,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum,V3a: tyop_2Enum_2Enum] :
( ( ( c_2Ereal_2Epow @ V1x @ c_2Enum_2E0 )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V2n ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V2n ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ V3a ) ) @ ( c_2Earithmetic_2ENUMERAL @ V2n ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ V3a ) @ ( c_2Earithmetic_2ENUMERAL @ V2n ) ) ) )
& ( ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ V3a ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ V2n ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) @ ( c_2Earithmetic_2EODD @ ( c_2Earithmetic_2ENUMERAL @ V2n ) ) @ c_2Erealax_2Ereal__neg
@ ^ [V4x: tyop_2Erealax_2Ereal] : V4x
@ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ V3a ) @ ( c_2Earithmetic_2ENUMERAL @ V2n ) ) ) ) )
& ( ( c_2Ereal_2Epow @ ( c_2Ereal_2E_2F @ V1x @ V0y ) @ V2n )
= ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ V1x @ V2n ) @ ( c_2Ereal_2Epow @ V0y @ V2n ) ) ) ) ).
thf(thm_2Ereal_2Eneg__rat,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ V1x @ V0y ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1x @ V0y ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__neg @ V1x ) @ V0y ) ) )
& ( ( c_2Ereal_2E_2F @ V1x @ ( c_2Erealax_2Ereal__neg @ V0y ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1x @ V0y ) ) @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__neg @ V1x ) @ V0y ) ) ) ) ).
thf(thm_2Ereal_2Eeq__rat,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2v: tyop_2Erealax_2Ereal,V3u: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2E_2F @ V1x @ V0y )
= ( c_2Ereal_2E_2F @ V3u @ V2v ) )
<=> ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1x @ V0y ) ) @ ( c_2Ereal_2E_2F @ V3u @ V2v ) ) @ ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V2v @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1x @ V0y ) @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V3u @ V2v ) ) ) @ ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0y @ V2v ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V1x @ V3u ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V1x @ V2v ) @ ( c_2Erealax_2Ereal__mul @ V0y @ V3u ) ) ) ) ) ) ).
thf(thm_2Ereal_2Eeq__ratl,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2E_2F @ V2x @ V1y )
= V0z )
<=> ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V2x @ V1y ) ) @ V0z ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V2x @ ( c_2Erealax_2Ereal__mul @ V1y @ V0z ) ) ) ) ).
thf(thm_2Ereal_2Eeq__ratr,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( V0z
= ( c_2Ereal_2E_2F @ V2x @ V1y ) )
<=> ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0z @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V2x @ V1y ) ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V1y @ V0z ) @ V2x ) ) ) ).
thf(thm_2Ereal_2Eeq__ints,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( ( c_2Ereal_2Ereal__of__num @ V0n )
= ( c_2Ereal_2Ereal__of__num @ V1m ) )
<=> ( V0n = V1m ) )
& ( ( ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V0n ) )
= ( c_2Ereal_2Ereal__of__num @ V1m ) )
<=> ( ( V0n = c_2Enum_2E0 )
& ( V1m = c_2Enum_2E0 ) ) )
& ( ( ( c_2Ereal_2Ereal__of__num @ V0n )
= ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V1m ) ) )
<=> ( ( V0n = c_2Enum_2E0 )
& ( V1m = c_2Enum_2E0 ) ) )
& ( ( ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V0n ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V1m ) ) )
<=> ( V0n = V1m ) ) ) ).
thf(thm_2Ereal_2Ediv__ratr,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2E_2F @ V2x @ ( c_2Ereal_2E_2F @ V1y @ V0z ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0z @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) @ ( c_2Ereal_2E_2F @ V2x @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1y @ V0z ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__mul @ V2x @ V0z ) @ V1y ) ) ) ).
thf(thm_2Ereal_2Ediv__ratl,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2E_2F @ ( c_2Ereal_2E_2F @ V2x @ V1y ) @ V0z )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V1y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2E_2F @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V2x @ V1y ) ) @ V0z ) @ ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0z @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2E_2F @ V2x @ V1y ) @ V0z ) ) @ ( c_2Ereal_2E_2F @ V2x @ ( c_2Erealax_2Ereal__mul @ V1y @ V0z ) ) ) ) ) ).
thf(thm_2Ereal_2Ediv__rat,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2v: tyop_2Erealax_2Ereal,V3u: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2E_2F @ ( c_2Ereal_2E_2F @ V1x @ V0y ) @ ( c_2Ereal_2E_2F @ V3u @ V2v ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V3u @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V2v @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2E_2F @ V1x @ V0y ) @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V3u @ V2v ) ) ) @ ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2E_2F @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1x @ V0y ) ) @ ( c_2Ereal_2E_2F @ V3u @ V2v ) ) @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__mul @ V1x @ V2v ) @ ( c_2Erealax_2Ereal__mul @ V0y @ V3u ) ) ) ) ) ).
thf(thm_2Ereal_2Ele__rat,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1u: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ V2n ) ) @ ( c_2Ereal_2E_2F @ V1u @ ( c_2Ereal_2Ereal__of__num @ V3m ) ) )
= ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2n @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) @ ( c_2Ereal_2E_2F @ V1u @ ( c_2Ereal_2Ereal__of__num @ V3m ) ) ) @ ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V3m @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ V2n ) ) @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1u @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V3m ) @ V0x ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2n ) @ V1u ) ) ) ) ) ).
thf(thm_2Ereal_2Ele__ratl,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1u: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ V2n ) ) @ V1u )
= ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2n @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) @ V1u ) @ ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2n ) @ V1u ) ) ) ) ).
thf(thm_2Ereal_2Ele__ratr,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1u: tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2E_2F @ V1u @ ( c_2Ereal_2Ereal__of__num @ V2m ) ) )
= ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2m @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1u @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) @ ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2m ) @ V0x ) @ V1u ) ) ) ).
thf(thm_2Ereal_2Ele__int,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Ereal_2Ereal__of__num @ V1m ) )
= ( c_2Earithmetic_2E_3C_3D @ V0n @ V1m ) )
& ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) @ ( c_2Ereal_2Ereal__of__num @ V1m ) )
= c_2Ebool_2ET )
& ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V1m ) ) )
<=> ( ( V0n = c_2Enum_2E0 )
& ( V1m = c_2Enum_2E0 ) ) )
& ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V1m ) ) )
= ( c_2Earithmetic_2E_3C_3D @ V1m @ V0n ) ) ) ).
thf(thm_2Ereal_2Elt__rat,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1u: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ V2n ) ) @ ( c_2Ereal_2E_2F @ V1u @ ( c_2Ereal_2Ereal__of__num @ V3m ) ) )
= ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2n @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) @ ( c_2Ereal_2E_2F @ V1u @ ( c_2Ereal_2Ereal__of__num @ V3m ) ) ) @ ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V3m @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ V2n ) ) @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1u @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V3m ) @ V0x ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2n ) @ V1u ) ) ) ) ) ).
thf(thm_2Ereal_2Elt__ratl,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1u: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ V2n ) ) @ V1u )
= ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2n @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) @ V1u ) @ ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2n ) @ V1u ) ) ) ) ).
thf(thm_2Ereal_2Elt__ratr,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1u: tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2E_2F @ V1u @ ( c_2Ereal_2Ereal__of__num @ V2m ) ) )
= ( c_2Ebool_2ECOND @ $o @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2m @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Emarker_2Eunint @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ V1u @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2m ) @ V0x ) @ V1u ) ) ) ).
thf(thm_2Ereal_2Elt__int,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Ereal_2Ereal__of__num @ V1m ) )
= ( c_2Eprim__rec_2E_3C @ V0n @ V1m ) )
& ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) @ ( c_2Ereal_2Ereal__of__num @ V1m ) )
<=> ( ( (~) @ ( V0n = c_2Enum_2E0 ) )
| ( (~) @ ( V1m = c_2Enum_2E0 ) ) ) )
& ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V1m ) ) )
= c_2Ebool_2EF )
& ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V1m ) ) )
= ( c_2Eprim__rec_2E_3C @ V1m @ V0n ) ) ) ).
thf(thm_2Ereal_2ENUM__FLOOR__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Ereal_2ENUM__FLOOR @ V0x ) ) @ V0x ) ) ).
thf(thm_2Ereal_2ENUM__FLOOR__LE2,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1x: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0y )
=> ( ( c_2Earithmetic_2E_3C_3D @ V1x @ ( c_2Ereal_2ENUM__FLOOR @ V0y ) )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ V1x ) @ V0y ) ) ) ).
thf(thm_2Ereal_2ENUM__FLOOR__LET,axiom,
! [V0y: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Ereal_2ENUM__FLOOR @ V1x ) @ V0y )
= ( c_2Erealax_2Ereal__lt @ V1x @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ V0y ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ereal_2ENUM__FLOOR__DIV,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0y ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Ereal_2ENUM__FLOOR @ ( c_2Ereal_2E_2F @ V1x @ V0y ) ) ) @ V0y ) @ V1x ) ) ).
thf(thm_2Ereal_2ENUM__FLOOR__DIV__LOWERBOUND,axiom,
! [V0y: tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0y ) )
=> ( c_2Erealax_2Ereal__lt @ V1x @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2B @ ( c_2Ereal_2ENUM__FLOOR @ ( c_2Ereal_2E_2F @ V1x @ V0y ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0y ) ) ) ).
thf(thm_2Ereal_2ENUM__FLOOR__BASE,axiom,
! [V0r: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0r @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( ( c_2Ereal_2ENUM__FLOOR @ V0r )
= c_2Enum_2E0 ) ) ).
thf(thm_2Ereal_2ENUM__FLOOR__EQNS,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( ( c_2Ereal_2ENUM__FLOOR @ ( c_2Ereal_2Ereal__of__num @ V0n ) )
= V0n )
& ( ( c_2Ereal_2ENUM__FLOOR @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) )
= c_2Enum_2E0 )
& ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1m )
=> ( ( c_2Ereal_2ENUM__FLOOR @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Ereal_2Ereal__of__num @ V1m ) ) )
= ( c_2Earithmetic_2EDIV @ V0n @ V1m ) ) )
& ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1m )
=> ( ( c_2Ereal_2ENUM__FLOOR @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) @ ( c_2Ereal_2Ereal__of__num @ V1m ) ) )
= c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal_2ENUM__FLOOR__LOWER__BOUND,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
= ( c_2Earithmetic_2E_3C_3D @ ( c_2Ereal_2ENUM__FLOOR @ ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ V1n ) ) ).
thf(thm_2Ereal_2ENUM__FLOOR__upper__bound,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ V0x )
= ( c_2Eprim__rec_2E_3C @ V1n @ ( c_2Ereal_2ENUM__FLOOR @ ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Ereal_2ENUM__CEILING__NUM__FLOOR,axiom,
! [V0r: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2ENUM__CEILING @ V0r )
= ( c_2Ebool_2ELET @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2E_5C_2F @ ( c_2Ereal_2Ereal__lte @ V0r @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ V0r @ ( c_2Ereal_2Ereal__of__num @ V1n ) ) ) @ V1n @ ( c_2Earithmetic_2E_2B @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
@ ( c_2Ereal_2ENUM__FLOOR @ V0r ) ) ) ).
thf(thm_2Ereal_2ELE__NUM__CEILING,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Ereal_2ENUM__CEILING @ V0x ) ) ) ).
thf(thm_2Ereal_2ENUM__CEILING__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ V1n ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Ereal_2ENUM__CEILING @ V0x ) @ V1n ) ) ).
%------------------------------------------------------------------------------