ITP001 Axioms: ITP127^7.ax
%------------------------------------------------------------------------------
% File : ITP127^7 : 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 : seq.ax [Gau19]
% : HL4127^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 185 ( 28 unt; 68 typ; 0 def)
% Number of atoms : 361 ( 31 equ; 4 cnn)
% Maximal formula atoms : 11 ( 1 avg)
% Number of connectives : 1564 ( 4 ~; 6 |; 75 &;1356 @)
% ( 26 <=>; 97 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 9 avg;1356 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 276 ( 276 >; 0 *; 0 +; 0 <<)
% Number of symbols : 67 ( 65 usr; 4 con; 0-6 aty)
% Number of variables : 394 ( 41 ^ 303 !; 22 ?; 394 :)
% ( 28 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emetric_2Emetric,type,
tyop_2Emetric_2Emetric: $tType > $tType ).
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(tyop_2Etopology_2Etopology,type,
tyop_2Etopology_2Etopology: $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Eseq_2E_2D_2D_3E,type,
c_2Eseq_2E_2D_2D_3E: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2E_2F,type,
c_2Ereal_2E_2F: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_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_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Epred__set_2EBIJ,type,
c_2Epred__set_2EBIJ:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Epred__set_2ECROSS,type,
c_2Epred__set_2ECROSS:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ecombin_2EK,type,
c_2Ecombin_2EK:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > A_27a ) ).
thf(c_2Earithmetic_2EMAX,type,
c_2Earithmetic_2EMAX: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ereal_2Eabs,type,
c_2Ereal_2Eabs: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Enets_2Ebounded,type,
c_2Enets_2Ebounded:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ ( tyop_2Emetric_2Emetric @ A_27a ) @ ( A_27b > A_27b > $o ) ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Eseq_2Ecauchy,type,
c_2Eseq_2Ecauchy: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Eseq_2Econvergent,type,
c_2Eseq_2Econvergent: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Epred__set_2Ecount,type,
c_2Epred__set_2Ecount: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Erealax_2Einv,type,
c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Eseq_2Elim,type,
c_2Eseq_2Elim: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Eseq_2Emono,type,
c_2Eseq_2Emono: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Eseq_2Emono__decreasing,type,
c_2Eseq_2Emono__decreasing: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Eseq_2Emono__increasing,type,
c_2Eseq_2Emono__increasing: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Emetric_2Emr1,type,
c_2Emetric_2Emr1: tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ).
thf(c_2Emetric_2Emtop,type,
c_2Emetric_2Emtop:
!>[A_27a: $tType] : ( ( tyop_2Emetric_2Emetric @ A_27a ) > ( tyop_2Etopology_2Etopology @ A_27a ) ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Ereal_2Epow,type,
c_2Ereal_2Epow: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_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_2Eseq_2Esubseq,type,
c_2Eseq_2Esubseq: ( tyop_2Enum_2Enum > tyop_2Enum_2Enum ) > $o ).
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_2Eseq_2Esuminf,type,
c_2Eseq_2Esuminf: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Eseq_2Esummable,type,
c_2Eseq_2Esummable: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Eseq_2Esums,type,
c_2Eseq_2Esums: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2Esup,type,
c_2Ereal_2Esup: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal ).
thf(c_2Enets_2Etends,type,
c_2Enets_2Etends:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a ) > A_27a > ( tyop_2Epair_2Eprod @ ( tyop_2Etopology_2Etopology @ A_27a ) @ ( A_27b > A_27b > $o ) ) > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Eseq_2Etends__num__real,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2E_2D_2D_3E @ V0x @ V1x0 )
= ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ V0x @ V1x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ c_2Earithmetic_2E_3E_3D ) ) ) ).
thf(thm_2Eseq_2Econvergent,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Econvergent @ V0f )
<=> ? [V1l: tyop_2Erealax_2Ereal] : ( c_2Eseq_2E_2D_2D_3E @ V0f @ V1l ) ) ).
thf(thm_2Eseq_2Ecauchy,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Ecauchy @ V0f )
<=> ! [V1e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e )
=> ? [V2N: tyop_2Enum_2Enum] :
! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3E_3D @ V3m @ V2N )
& ( c_2Earithmetic_2E_3E_3D @ V4n @ V2N ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V0f @ V3m ) @ ( V0f @ V4n ) ) ) @ V1e ) ) ) ) ).
thf(thm_2Eseq_2Elim,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Elim @ V0f )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V1l: tyop_2Erealax_2Ereal] : ( c_2Eseq_2E_2D_2D_3E @ V0f @ V1l ) ) ) ).
thf(thm_2Eseq_2Esubseq,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Eseq_2Esubseq @ V0f )
<=> ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1m @ V2n )
=> ( c_2Eprim__rec_2E_3C @ ( V0f @ V1m ) @ ( V0f @ V2n ) ) ) ) ).
thf(thm_2Eseq_2Emono,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Emono @ V0f )
<=> ( ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V1m ) @ ( V0f @ V2n ) ) )
| ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V3m @ V4n )
=> ( c_2Ereal_2Ereal__ge @ ( V0f @ V3m ) @ ( V0f @ V4n ) ) ) ) ) ).
thf(thm_2Eseq_2Esums,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esums @ V0f @ V1s )
= ( c_2Eseq_2E_2D_2D_3E
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V2n ) @ V0f )
@ V1s ) ) ).
thf(thm_2Eseq_2Esummable,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esummable @ V0f )
<=> ? [V1s: tyop_2Erealax_2Ereal] : ( c_2Eseq_2Esums @ V0f @ V1s ) ) ).
thf(thm_2Eseq_2Esuminf,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esuminf @ V0f )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V1s: tyop_2Erealax_2Ereal] : ( c_2Eseq_2Esums @ V0f @ V1s ) ) ) ).
thf(thm_2Eseq_2Emono__increasing__def,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Emono__increasing @ V0f )
<=> ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V1m ) @ ( V0f @ V2n ) ) ) ) ).
thf(thm_2Eseq_2Emono__decreasing__def,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Emono__decreasing @ V0f )
<=> ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V2n ) @ ( V0f @ V1m ) ) ) ) ).
thf(thm_2Eseq_2ESEQ,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2E_2D_2D_3E @ V0x @ V1x0 )
<=> ! [V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ? [V3N: tyop_2Enum_2Enum] :
! [V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V4n @ V3N )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( V0x @ V4n ) @ V1x0 ) ) @ V2e ) ) ) ) ).
thf(thm_2Eseq_2ESEQ__CONST,axiom,
! [V0k: tyop_2Erealax_2Ereal] :
( c_2Eseq_2E_2D_2D_3E
@ ^ [V1x: tyop_2Enum_2Enum] : V0k
@ V0k ) ).
thf(thm_2Eseq_2ESEQ__ADD,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3y0: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2E_2D_2D_3E @ V0x @ V1x0 )
& ( c_2Eseq_2E_2D_2D_3E @ V2y @ V3y0 ) )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add @ ( V0x @ V4n ) @ ( V2y @ V4n ) )
@ ( c_2Erealax_2Ereal__add @ V1x0 @ V3y0 ) ) ) ).
thf(thm_2Eseq_2ESEQ__MUL,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3y0: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2E_2D_2D_3E @ V0x @ V1x0 )
& ( c_2Eseq_2E_2D_2D_3E @ V2y @ V3y0 ) )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0x @ V4n ) @ ( V2y @ V4n ) )
@ ( c_2Erealax_2Ereal__mul @ V1x0 @ V3y0 ) ) ) ).
thf(thm_2Eseq_2ESEQ__NEG,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2E_2D_2D_3E @ V0x @ V1x0 )
= ( c_2Eseq_2E_2D_2D_3E
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( V0x @ V2n ) )
@ ( c_2Erealax_2Ereal__neg @ V1x0 ) ) ) ).
thf(thm_2Eseq_2ESEQ__INV,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2E_2D_2D_3E @ V0x @ V1x0 )
& ( (~)
@ ( V1x0
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Einv @ ( V0x @ V2n ) )
@ ( c_2Erealax_2Einv @ V1x0 ) ) ) ).
thf(thm_2Eseq_2ESEQ__SUB,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3y0: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2E_2D_2D_3E @ V0x @ V1x0 )
& ( c_2Eseq_2E_2D_2D_3E @ V2y @ V3y0 ) )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0x @ V4n ) @ ( V2y @ V4n ) )
@ ( c_2Ereal_2Ereal__sub @ V1x0 @ V3y0 ) ) ) ).
thf(thm_2Eseq_2ESEQ__DIV,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3y0: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2E_2D_2D_3E @ V0x @ V1x0 )
& ( c_2Eseq_2E_2D_2D_3E @ V2y @ V3y0 )
& ( (~)
@ ( V3y0
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2E_2F @ ( V0x @ V4n ) @ ( V2y @ V4n ) )
@ ( c_2Ereal_2E_2F @ V1x0 @ V3y0 ) ) ) ).
thf(thm_2Eseq_2ESEQ__UNIQ,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x1: tyop_2Erealax_2Ereal,V2x2: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2E_2D_2D_3E @ V0x @ V1x1 )
& ( c_2Eseq_2E_2D_2D_3E @ V0x @ V2x2 ) )
=> ( V1x1 = V2x2 ) ) ).
thf(thm_2Eseq_2ESEQ__LIM,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Econvergent @ V0f )
= ( c_2Eseq_2E_2D_2D_3E @ V0f @ ( c_2Eseq_2Elim @ V0f ) ) ) ).
thf(thm_2Eseq_2ESUBSEQ__SUC,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Eseq_2Esubseq @ V0f )
<=> ! [V1n: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C @ ( V0f @ V1n ) @ ( V0f @ ( c_2Enum_2ESUC @ V1n ) ) ) ) ).
thf(thm_2Eseq_2EMONO__SUC,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Emono @ V0f )
<=> ( ! [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__ge @ ( V0f @ ( c_2Enum_2ESUC @ V1n ) ) @ ( V0f @ V1n ) )
| ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ ( c_2Enum_2ESUC @ V2n ) ) @ ( V0f @ V2n ) ) ) ) ).
thf(thm_2Eseq_2EMAX__LEMMA,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1N: tyop_2Enum_2Enum] :
? [V2k: tyop_2Erealax_2Ereal] :
! [V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3n @ V1N )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( V0s @ V3n ) ) @ V2k ) ) ).
thf(thm_2Eseq_2ESEQ__BOUNDED,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ) @ c_2Emetric_2Emr1 @ c_2Earithmetic_2E_3E_3D ) @ V0s )
<=> ? [V1k: tyop_2Erealax_2Ereal] :
! [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( V0s @ V2n ) ) @ V1k ) ) ).
thf(thm_2Eseq_2ESEQ__BOUNDED__2,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k: tyop_2Erealax_2Ereal,V2k_27: tyop_2Erealax_2Ereal] :
( ! [V3n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ V1k @ ( V0f @ V3n ) )
& ( c_2Ereal_2Ereal__lte @ ( V0f @ V3n ) @ V2k_27 ) )
=> ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ) @ c_2Emetric_2Emr1 @ c_2Earithmetic_2E_3E_3D ) @ V0f ) ) ).
thf(thm_2Eseq_2ESEQ__CBOUNDED,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Ecauchy @ V0f )
=> ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ) @ c_2Emetric_2Emr1 @ c_2Earithmetic_2E_3E_3D ) @ V0f ) ) ).
thf(thm_2Eseq_2ESEQ__ICONV,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ) @ c_2Emetric_2Emr1 @ c_2Earithmetic_2E_3E_3D ) @ V0f )
& ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V1m @ V2n )
=> ( c_2Ereal_2Ereal__ge @ ( V0f @ V1m ) @ ( V0f @ V2n ) ) ) )
=> ( c_2Eseq_2Econvergent @ V0f ) ) ).
thf(thm_2Eseq_2ESEQ__NEG__CONV,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Econvergent @ V0f )
= ( c_2Eseq_2Econvergent
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( V0f @ V1n ) ) ) ) ).
thf(thm_2Eseq_2ESEQ__NEG__BOUNDED,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ) @ c_2Emetric_2Emr1 @ c_2Earithmetic_2E_3E_3D )
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( V0f @ V1n ) ) )
= ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ) @ c_2Emetric_2Emr1 @ c_2Earithmetic_2E_3E_3D ) @ V0f ) ) ).
thf(thm_2Eseq_2ESEQ__BCONV,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ) @ c_2Emetric_2Emr1 @ c_2Earithmetic_2E_3E_3D ) @ V0f )
& ( c_2Eseq_2Emono @ V0f ) )
=> ( c_2Eseq_2Econvergent @ V0f ) ) ).
thf(thm_2Eseq_2ESEQ__MONOSUB,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
? [V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Eseq_2Esubseq @ V1f )
& ( c_2Eseq_2Emono
@ ^ [V2n: tyop_2Enum_2Enum] : ( V0s @ ( V1f @ V2n ) ) ) ) ).
thf(thm_2Eseq_2ESEQ__SBOUNDED,axiom,
! [V0s: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ) @ c_2Emetric_2Emr1 @ c_2Earithmetic_2E_3E_3D ) @ V0s )
=> ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ) @ c_2Emetric_2Emr1 @ c_2Earithmetic_2E_3E_3D )
@ ^ [V2n: tyop_2Enum_2Enum] : ( V0s @ ( V1f @ V2n ) ) ) ) ).
thf(thm_2Eseq_2ESEQ__SUBLE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Eseq_2Esubseq @ V0f )
=> ! [V1n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D @ V1n @ ( V0f @ V1n ) ) ) ).
thf(thm_2Eseq_2ESEQ__DIRECT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Eseq_2Esubseq @ V0f )
=> ! [V1N1: tyop_2Enum_2Enum,V2N2: tyop_2Enum_2Enum] :
? [V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V3n @ V1N1 )
& ( c_2Earithmetic_2E_3E_3D @ ( V0f @ V3n ) @ V2N2 ) ) ) ).
thf(thm_2Eseq_2ESEQ__CAUCHY,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Ecauchy @ V0f )
= ( c_2Eseq_2Econvergent @ V0f ) ) ).
thf(thm_2Eseq_2ESEQ__LE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2l: tyop_2Erealax_2Ereal,V3m: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2E_2D_2D_3E @ V0f @ V2l )
& ( c_2Eseq_2E_2D_2D_3E @ V1g @ V3m )
& ? [V4N: tyop_2Enum_2Enum] :
! [V5n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V5n @ V4N )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V5n ) @ ( V1g @ V5n ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ V2l @ V3m ) ) ).
thf(thm_2Eseq_2ESEQ__SUC,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2E_2D_2D_3E @ V0f @ V1l )
= ( c_2Eseq_2E_2D_2D_3E
@ ^ [V2n: tyop_2Enum_2Enum] : ( V0f @ ( c_2Enum_2ESUC @ V2n ) )
@ V1l ) ) ).
thf(thm_2Eseq_2ESEQ__ABS,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2E_2D_2D_3E
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0f @ V1n ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Eseq_2E_2D_2D_3E @ V0f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eseq_2ESEQ__ABS__IMP,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2E_2D_2D_3E @ V0f @ V1l )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0f @ V2n ) )
@ ( c_2Ereal_2Eabs @ V1l ) ) ) ).
thf(thm_2Eseq_2ESEQ__INV0,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ! [V1y: tyop_2Erealax_2Ereal] :
? [V2N: tyop_2Enum_2Enum] :
! [V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V3n @ V2N )
=> ( c_2Ereal_2Ereal__gt @ ( V0f @ V3n ) @ V1y ) )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Einv @ ( V0f @ V4n ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eseq_2ESEQ__POWER__ABS,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V0c ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow @ ( c_2Ereal_2Eabs @ V0c ) @ V1n )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eseq_2ESEQ__POWER,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V0c ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow @ V0c @ V1n )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eseq_2ENEST__LEMMA,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__ge @ ( V0f @ ( c_2Enum_2ESUC @ V2n ) ) @ ( V0f @ V2n ) )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V1g @ ( c_2Enum_2ESUC @ V3n ) ) @ ( V1g @ V3n ) )
& ! [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ V4n ) @ ( V1g @ V4n ) ) )
=> ? [V5l: tyop_2Erealax_2Ereal,V6m: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V5l @ V6m )
& ! [V7n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ V7n ) @ V5l )
& ( c_2Eseq_2E_2D_2D_3E @ V0f @ V5l )
& ! [V8n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ V6m @ ( V1g @ V8n ) )
& ( c_2Eseq_2E_2D_2D_3E @ V1g @ V6m ) ) ) ).
thf(thm_2Eseq_2ENEST__LEMMA__UNIQ,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__ge @ ( V0f @ ( c_2Enum_2ESUC @ V2n ) ) @ ( V0f @ V2n ) )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V1g @ ( c_2Enum_2ESUC @ V3n ) ) @ ( V1g @ V3n ) )
& ! [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ V4n ) @ ( V1g @ V4n ) )
& ( c_2Eseq_2E_2D_2D_3E
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0f @ V5n ) @ ( V1g @ V5n ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ? [V6l: tyop_2Erealax_2Ereal] :
( ! [V7n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ V7n ) @ V6l )
& ( c_2Eseq_2E_2D_2D_3E @ V0f @ V6l )
& ! [V8n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ V6l @ ( V1g @ V8n ) )
& ( c_2Eseq_2E_2D_2D_3E @ V1g @ V6l ) ) ) ).
thf(thm_2Eseq_2EBOLZANO__LEMMA,axiom,
! [V0P: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > $o] :
( ( ! [V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal,V3c: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V1a @ V2b )
& ( c_2Ereal_2Ereal__lte @ V2b @ V3c )
& ( V0P @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) )
& ( V0P @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2b @ V3c ) ) )
=> ( V0P @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V3c ) ) )
& ! [V4x: tyop_2Erealax_2Ereal] :
? [V5d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5d )
& ! [V6a: tyop_2Erealax_2Ereal,V7b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V6a @ V4x )
& ( c_2Ereal_2Ereal__lte @ V4x @ V7b )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__sub @ V7b @ V6a ) @ V5d ) )
=> ( V0P @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V6a @ V7b ) ) ) ) )
=> ! [V8a: tyop_2Erealax_2Ereal,V9b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V8a @ V9b )
=> ( V0P @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V8a @ V9b ) ) ) ) ).
thf(thm_2Eseq_2ESUM__SUMMABLE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esums @ V0f @ V1l )
=> ( c_2Eseq_2Esummable @ V0f ) ) ).
thf(thm_2Eseq_2ESUMMABLE__SUM,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esummable @ V0f )
=> ( c_2Eseq_2Esums @ V0f @ ( c_2Eseq_2Esuminf @ V0f ) ) ) ).
thf(thm_2Eseq_2ESUM__UNIQ,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esums @ V0f @ V1x )
=> ( V1x
= ( c_2Eseq_2Esuminf @ V0f ) ) ) ).
thf(thm_2Eseq_2ESER__0,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ! [V2m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1n @ V2m )
=> ( ( V0f @ V2m )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( c_2Eseq_2Esums @ V0f @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1n ) @ V0f ) ) ) ).
thf(thm_2Eseq_2ESER__POS__LE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eseq_2Esummable @ V0f )
& ! [V2m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1n @ V2m )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V2m ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1n ) @ V0f ) @ ( c_2Eseq_2Esuminf @ V0f ) ) ) ).
thf(thm_2Eseq_2ESER__POS__LT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eseq_2Esummable @ V0f )
& ! [V2m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1n @ V2m )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V2m ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1n ) @ V0f ) @ ( c_2Eseq_2Esuminf @ V0f ) ) ) ).
thf(thm_2Eseq_2ESER__GROUP,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k: tyop_2Enum_2Enum] :
( ( ( c_2Eseq_2Esummable @ V0f )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1k ) )
=> ( c_2Eseq_2Esums
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_2A @ V2n @ V1k ) @ V1k ) @ V0f )
@ ( c_2Eseq_2Esuminf @ V0f ) ) ) ).
thf(thm_2Eseq_2ESER__PAIR,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esummable @ V0f )
=> ( c_2Eseq_2Esums
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0f )
@ ( c_2Eseq_2Esuminf @ V0f ) ) ) ).
thf(thm_2Eseq_2ESER__OFFSET,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esummable @ V0f )
=> ! [V1k: tyop_2Enum_2Enum] :
( c_2Eseq_2Esums
@ ^ [V2n: tyop_2Enum_2Enum] : ( V0f @ ( c_2Earithmetic_2E_2B @ V2n @ V1k ) )
@ ( c_2Ereal_2Ereal__sub @ ( c_2Eseq_2Esuminf @ V0f ) @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1k ) @ V0f ) ) ) ) ).
thf(thm_2Eseq_2ESER__POS__LT__PAIR,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eseq_2Esummable @ V0f )
& ! [V2d: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Erealax_2Ereal__add @ ( V0f @ ( c_2Earithmetic_2E_2B @ V1n @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2d ) ) ) @ ( V0f @ ( c_2Earithmetic_2E_2B @ V1n @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2d ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1n ) @ V0f ) @ ( c_2Eseq_2Esuminf @ V0f ) ) ) ).
thf(thm_2Eseq_2ESER__ADD,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3y0: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2Esums @ V0x @ V1x0 )
& ( c_2Eseq_2Esums @ V2y @ V3y0 ) )
=> ( c_2Eseq_2Esums
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add @ ( V0x @ V4n ) @ ( V2y @ V4n ) )
@ ( c_2Erealax_2Ereal__add @ V1x0 @ V3y0 ) ) ) ).
thf(thm_2Eseq_2ESER__CMUL,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esums @ V0x @ V1x0 )
=> ( c_2Eseq_2Esums
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ V2c @ ( V0x @ V3n ) )
@ ( c_2Erealax_2Ereal__mul @ V2c @ V1x0 ) ) ) ).
thf(thm_2Eseq_2ESER__NEG,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esums @ V0x @ V1x0 )
=> ( c_2Eseq_2Esums
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( V0x @ V2n ) )
@ ( c_2Erealax_2Ereal__neg @ V1x0 ) ) ) ).
thf(thm_2Eseq_2ESER__SUB,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3y0: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2Esums @ V0x @ V1x0 )
& ( c_2Eseq_2Esums @ V2y @ V3y0 ) )
=> ( c_2Eseq_2Esums
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0x @ V4n ) @ ( V2y @ V4n ) )
@ ( c_2Ereal_2Ereal__sub @ V1x0 @ V3y0 ) ) ) ).
thf(thm_2Eseq_2ESER__CDIV,axiom,
! [V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x0: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esums @ V0x @ V1x0 )
=> ( c_2Eseq_2Esums
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2E_2F @ ( V0x @ V3n ) @ V2c )
@ ( c_2Ereal_2E_2F @ V1x0 @ V2c ) ) ) ).
thf(thm_2Eseq_2ESER__CAUCHY,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esummable @ V0f )
<=> ! [V1e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1e )
=> ? [V2N: tyop_2Enum_2Enum] :
! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V3m @ V2N )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V3m @ V4n ) @ V0f ) ) @ V1e ) ) ) ) ).
thf(thm_2Eseq_2ESER__ZERO,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esummable @ V0f )
=> ( c_2Eseq_2E_2D_2D_3E @ V0f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eseq_2ESER__COMPAR,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ? [V2N: tyop_2Enum_2Enum] :
! [V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V3n @ V2N )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V3n ) ) @ ( V1g @ V3n ) ) )
& ( c_2Eseq_2Esummable @ V1g ) )
=> ( c_2Eseq_2Esummable @ V0f ) ) ).
thf(thm_2Eseq_2ESER__COMPARA,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ? [V2N: tyop_2Enum_2Enum] :
! [V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V3n @ V2N )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V3n ) ) @ ( V1g @ V3n ) ) )
& ( c_2Eseq_2Esummable @ V1g ) )
=> ( c_2Eseq_2Esummable
@ ^ [V4k: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0f @ V4k ) ) ) ) ).
thf(thm_2Eseq_2ESER__LE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ V2n ) @ ( V1g @ V2n ) )
& ( c_2Eseq_2Esummable @ V0f )
& ( c_2Eseq_2Esummable @ V1g ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eseq_2Esuminf @ V0f ) @ ( c_2Eseq_2Esuminf @ V1g ) ) ) ).
thf(thm_2Eseq_2ESER__LE2,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V2n ) ) @ ( V1g @ V2n ) )
& ( c_2Eseq_2Esummable @ V1g ) )
=> ( ( c_2Eseq_2Esummable @ V0f )
& ( c_2Ereal_2Ereal__lte @ ( c_2Eseq_2Esuminf @ V0f ) @ ( c_2Eseq_2Esuminf @ V1g ) ) ) ) ).
thf(thm_2Eseq_2ESER__ACONV,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esummable
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0f @ V1n ) ) )
=> ( c_2Eseq_2Esummable @ V0f ) ) ).
thf(thm_2Eseq_2ESER__ABS,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esummable
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0f @ V1n ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Eseq_2Esuminf @ V0f ) )
@ ( c_2Eseq_2Esuminf
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0f @ V2n ) ) ) ) ) ).
thf(thm_2Eseq_2EGP__FINITE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1n )
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow @ V0x @ V2n ) )
= ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Epow @ V0x @ V1n ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__sub @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Eseq_2EGP,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( c_2Eseq_2Esums
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow @ V0x @ V1n )
@ ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) ) ) ) ).
thf(thm_2Eseq_2EABS__NEG__LEMMA,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0c @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
=> ! [V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V1x ) @ ( c_2Erealax_2Ereal__mul @ V0c @ ( c_2Ereal_2Eabs @ V2y ) ) )
=> ( V1x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eseq_2ESER__RATIO,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2N: tyop_2Enum_2Enum] :
( ( ( c_2Erealax_2Ereal__lt @ V1c @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V3n @ V2N )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ ( c_2Enum_2ESUC @ V3n ) ) ) @ ( c_2Erealax_2Ereal__mul @ V1c @ ( c_2Ereal_2Eabs @ ( V0f @ V3n ) ) ) ) ) )
=> ( c_2Eseq_2Esummable @ V0f ) ) ).
thf(thm_2Eseq_2ELE__SEQ__IMP__LE__LIM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ V0x @ ( V2f @ V3n ) )
& ( c_2Eseq_2E_2D_2D_3E @ V2f @ V1y ) )
=> ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ).
thf(thm_2Eseq_2ESEQ__LE__IMP__LIM__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V2f @ V3n ) @ V0x )
& ( c_2Eseq_2E_2D_2D_3E @ V2f @ V1y ) )
=> ( c_2Ereal_2Ereal__lte @ V1y @ V0x ) ) ).
thf(thm_2Eseq_2ESEQ__MONO__LE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( ! [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ V3n ) @ ( V0f @ ( c_2Earithmetic_2E_2B @ V3n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( c_2Eseq_2E_2D_2D_3E @ V0f @ V1x ) )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V2n ) @ V1x ) ) ).
thf(thm_2Eseq_2ESEQ__LE__MONO,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( ! [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ ( c_2Earithmetic_2E_2B @ V3n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V0f @ V3n ) )
& ( c_2Eseq_2E_2D_2D_3E @ V0f @ V1x ) )
=> ( c_2Ereal_2Ereal__lte @ V1x @ ( V0f @ V2n ) ) ) ).
thf(thm_2Eseq_2Emono__increasing__suc,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Emono__increasing @ V0f )
<=> ! [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ V1n ) @ ( V0f @ ( c_2Enum_2ESUC @ V1n ) ) ) ) ).
thf(thm_2Eseq_2Emono__decreasing__suc,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Emono__decreasing @ V0f )
<=> ! [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ ( c_2Enum_2ESUC @ V1n ) ) @ ( V0f @ V1n ) ) ) ).
thf(thm_2Eseq_2Emono__increasing__converges__to__sup,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1r: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2Emono__increasing @ V0f )
& ( c_2Eseq_2E_2D_2D_3E @ V0f @ V1r ) )
=> ( V1r
= ( c_2Ereal_2Esup @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ V0f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ).
thf(thm_2Eseq_2EINCREASING__SEQ,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal] :
( ( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ V2n ) @ ( V0f @ ( c_2Enum_2ESUC @ V2n ) ) )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ V3n ) @ V1l )
& ! [V4e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4e )
=> ? [V5n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ V1l @ ( c_2Erealax_2Ereal__add @ ( V0f @ V5n ) @ V4e ) ) ) )
=> ( c_2Eseq_2E_2D_2D_3E @ V0f @ V1l ) ) ).
thf(thm_2Eseq_2EMAX__LE__X,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2k: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EMAX @ V0m @ V1n ) @ V2k )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V2k )
& ( c_2Earithmetic_2E_3C_3D @ V1n @ V2k ) ) ) ).
thf(thm_2Eseq_2EX__LE__MAX,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2k: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V2k @ ( c_2Earithmetic_2EMAX @ V0m @ V1n ) )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V2k @ V0m )
| ( c_2Earithmetic_2E_3C_3D @ V2k @ V1n ) ) ) ).
thf(thm_2Eseq_2ETRANSFORM__2D__NUM,axiom,
! [V0P: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( ( ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( V0P @ V1m @ V2n )
=> ( V0P @ V2n @ V1m ) )
& ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] : ( V0P @ V3m @ ( c_2Earithmetic_2E_2B @ V3m @ V4n ) ) )
=> ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] : ( V0P @ V5m @ V6n ) ) ).
thf(thm_2Eseq_2ETRIANGLE__2D__NUM,axiom,
! [V0P: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
( ! [V1d: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] : ( V0P @ V2n @ ( c_2Earithmetic_2E_2B @ V1d @ V2n ) )
=> ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V3m @ V4n )
=> ( V0P @ V3m @ V4n ) ) ) ).
thf(thm_2Eseq_2ESEQ__SANDWICH,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2h: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2E_2D_2D_3E @ V0f @ V3l )
& ( c_2Eseq_2E_2D_2D_3E @ V2h @ V3l )
& ! [V4n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ ( V0f @ V4n ) @ ( V1g @ V4n ) )
& ( c_2Ereal_2Ereal__lte @ ( V1g @ V4n ) @ ( V2h @ V4n ) ) ) )
=> ( c_2Eseq_2E_2D_2D_3E @ V1g @ V3l ) ) ).
thf(thm_2Eseq_2ESER__POS,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2Esummable @ V0f )
& ! [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V1n ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eseq_2Esuminf @ V0f ) ) ) ).
thf(thm_2Eseq_2ESER__POS__MONO,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 ) )
=> ( c_2Eseq_2Emono
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V2n ) @ V0f ) ) ) ).
thf(thm_2Eseq_2EPOS__SUMMABLE,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 ) )
& ? [V2x: tyop_2Erealax_2Ereal] :
! [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V3n ) @ V0f ) @ V2x ) )
=> ( c_2Eseq_2Esummable @ V0f ) ) ).
thf(thm_2Eseq_2ESUMMABLE__LE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2Esummable @ V0f )
& ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V2n ) @ V0f ) @ V1x ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eseq_2Esuminf @ V0f ) @ V1x ) ) ).
thf(thm_2Eseq_2ESUMS__EQ,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esums @ V0f @ V1x )
<=> ( ( c_2Eseq_2Esummable @ V0f )
& ( ( c_2Eseq_2Esuminf @ V0f )
= V1x ) ) ) ).
thf(thm_2Eseq_2ESUMINF__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 ) )
& ( c_2Eseq_2Esummable @ V0f ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eseq_2Esuminf @ V0f ) ) ) ).
thf(thm_2Eseq_2ESUM__PICK,axiom,
! [V0n: tyop_2Enum_2Enum,V1k: tyop_2Enum_2Enum,V2x: 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_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V3m @ V1k ) @ V2x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Eprim__rec_2E_3C @ V1k @ V0n ) @ V2x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eseq_2ESUM__LT,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_2Erealax_2Ereal__lt @ ( V0f @ V4r ) @ ( V1g @ V4r ) ) )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V3n ) )
=> ( c_2Erealax_2Ereal__lt @ ( 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_2Eseq_2ESUM__CONST__R,axiom,
! [V0n: tyop_2Enum_2Enum,V1r: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0n ) @ ( c_2Ecombin_2EK @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ V1r ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ V1r ) ) ).
thf(thm_2Eseq_2ESUMS__ZERO,axiom,
c_2Eseq_2Esums @ ( c_2Ecombin_2EK @ tyop_2Erealax_2Ereal @ tyop_2Enum_2Enum @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ).
thf(thm_2Eseq_2ELT__SUC,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0a @ ( c_2Enum_2ESUC @ V1b ) )
<=> ( ( c_2Eprim__rec_2E_3C @ V0a @ V1b )
| ( V0a = V1b ) ) ) ).
thf(thm_2Eseq_2ELE__SUC,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0a @ ( c_2Enum_2ESUC @ V1b ) )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V0a @ V1b )
| ( V0a
= ( c_2Enum_2ESUC @ V1b ) ) ) ) ).
thf(thm_2Eseq_2EK__PARTIAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a] :
( ( c_2Ecombin_2EK @ A_27a @ A_27b @ V0x )
= ( ^ [V1z: A_27b] : V0x ) ) ).
thf(thm_2Eseq_2EHALF__POS,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 ) ) ) ) ).
thf(thm_2Eseq_2EHALF__LT__1,axiom,
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 ) ) ) ).
thf(thm_2Eseq_2EHALF__CANCEL,axiom,
( ( c_2Erealax_2Ereal__mul @ ( 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_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_2Eseq_2EX__HALF__HALF,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( 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 ) ) ) ) @ V0x ) @ ( c_2Erealax_2Ereal__mul @ ( 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 ) ) ) ) @ V0x ) )
= V0x ) ).
thf(thm_2Eseq_2EONE__MINUS__HALF,axiom,
( ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( 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_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 ) ) ) ) ) ).
thf(thm_2Eseq_2ENUM__2D__BIJ__BIG__SQUARE,axiom,
! [V0f: tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ),V1N: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V0f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
=> ? [V2k: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V0f @ ( c_2Epred__set_2Ecount @ V1N ) ) @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2Ecount @ V2k ) @ ( c_2Epred__set_2Ecount @ V2k ) ) ) ) ).
thf(thm_2Eseq_2ENUM__2D__BIJ__SMALL__SQUARE,axiom,
! [V0f: tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ),V1k: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V0f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
=> ? [V2N: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2Ecount @ V1k ) @ ( c_2Epred__set_2Ecount @ V1k ) ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V0f @ ( c_2Epred__set_2Ecount @ V2N ) ) ) ) ).
thf(thm_2Eseq_2ESUMINF__ADD,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2Esummable @ V0f )
& ( c_2Eseq_2Esummable @ V1g ) )
=> ( ( c_2Eseq_2Esummable
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add @ ( V0f @ V2n ) @ ( V1g @ V2n ) ) )
& ( ( c_2Erealax_2Ereal__add @ ( c_2Eseq_2Esuminf @ V0f ) @ ( c_2Eseq_2Esuminf @ V1g ) )
= ( c_2Eseq_2Esuminf
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add @ ( V0f @ V3n ) @ ( V1g @ V3n ) ) ) ) ) ) ).
thf(thm_2Eseq_2ESUMINF__2D,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2h: tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum )] :
( ( ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V3m @ V4n ) )
& ! [V5n: tyop_2Enum_2Enum] : ( c_2Eseq_2Esums @ ( V0f @ V5n ) @ ( V1g @ V5n ) )
& ( c_2Eseq_2Esummable @ V1g )
& ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ V2h @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) )
=> ( c_2Eseq_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ ( c_2Epair_2EUNCURRY @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ V0f ) @ V2h ) @ ( c_2Eseq_2Esuminf @ V1g ) ) ) ).
thf(thm_2Eseq_2EPOW__HALF__SER,axiom,
( c_2Eseq_2Esums
@ ^ [V0n: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow @ ( 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_2Earithmetic_2E_2B @ V0n @ ( 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_2Eseq_2ESER__POS__COMPARE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V2n ) )
& ( c_2Eseq_2Esummable @ V1g )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0f @ V3n ) @ ( V1g @ V3n ) ) )
=> ( ( c_2Eseq_2Esummable @ V0f )
& ( c_2Ereal_2Ereal__lte @ ( c_2Eseq_2Esuminf @ V0f ) @ ( c_2Eseq_2Esuminf @ V1g ) ) ) ) ).
%------------------------------------------------------------------------------