ITP001 Axioms: ITP132^7.ax
%------------------------------------------------------------------------------
% File : ITP132^7 : 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 : transc.ax [Gau19]
% : HL4132^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 306 ( 86 unt; 66 typ; 0 def)
% Number of atoms : 681 ( 190 equ; 23 cnn)
% Maximal formula atoms : 28 ( 2 avg)
% Number of connectives : 3889 ( 23 ~; 3 |; 156 &;3534 @)
% ( 20 <=>; 153 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg;3534 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 222 ( 222 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 63 usr; 4 con; 0-4 aty)
% Number of variables : 509 ( 65 ^ 409 !; 27 ?; 509 :)
% ( 8 !>; 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_2Etransc_2EDint,type,
c_2Etransc_2EDint: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Earithmetic_2EEVEN,type,
c_2Earithmetic_2EEVEN: tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2EFACT,type,
c_2Earithmetic_2EFACT: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_2Etransc_2Eacs,type,
c_2Etransc_2Eacs: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Easn,type,
c_2Etransc_2Easn: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Eatn,type,
c_2Etransc_2Eatn: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Ecos,type,
c_2Etransc_2Ecos: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Elim_2Ediffl,type,
c_2Elim_2Ediffl: ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Epowser_2Ediffs,type,
c_2Epowser_2Ediffs: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Edivision,type,
c_2Etransc_2Edivision: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Etransc_2Edsize,type,
c_2Etransc_2Edsize: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Enum_2Enum ).
thf(c_2Etransc_2Eexp,type,
c_2Etransc_2Eexp: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Efine,type,
c_2Etransc_2Efine: ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) > $o ).
thf(c_2Etransc_2Egauge,type,
c_2Etransc_2Egauge: ( tyop_2Erealax_2Ereal > $o ) > ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Erealax_2Einv,type,
c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Eln,type,
c_2Etransc_2Eln: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Epi,type,
c_2Etransc_2Epi: tyop_2Erealax_2Ereal ).
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_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_2Etransc_2Eroot,type,
c_2Etransc_2Eroot: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Erpow,type,
c_2Etransc_2Erpow: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Ersum,type,
c_2Etransc_2Ersum: ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) > ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Esin,type,
c_2Etransc_2Esin: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Esqrt,type,
c_2Etransc_2Esqrt: 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_2Eseq_2Esuminf,type,
c_2Eseq_2Esuminf: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Eseq_2Esums,type,
c_2Eseq_2Esums: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Etransc_2Etan,type,
c_2Etransc_2Etan: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Etdiv,type,
c_2Etransc_2Etdiv: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) > $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_2Etransc_2Eatn,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eatn @ V0y )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ V1x ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ V1x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Etan @ V1x ) @ V0y ) ) ) ) ) ).
thf(thm_2Etransc_2Eacs,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eacs @ V0y )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V1x @ c_2Etransc_2Epi ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V1x ) @ V0y ) ) ) ) ) ).
thf(thm_2Etransc_2Easn,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Easn @ V0y )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ V1x ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V1x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Esin @ V1x ) @ V0y ) ) ) ) ) ).
thf(thm_2Etransc_2Etan,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Etan @ V0x )
= ( c_2Ereal_2E_2F @ ( c_2Etransc_2Esin @ V0x ) @ ( c_2Etransc_2Ecos @ V0x ) ) ) ).
thf(thm_2Etransc_2Epi,axiom,
( c_2Etransc_2Epi
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
@ ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2Esqrt,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Esqrt @ V0x )
= ( c_2Etransc_2Eroot @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V0x ) ) ).
thf(thm_2Etransc_2Eroot,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eroot @ V0n @ V1x )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V2u: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D_3D_3E @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x ) @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2u ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Epow @ V2u @ V0n ) @ V1x ) ) ) ) ).
thf(thm_2Etransc_2Eexp,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eexp @ V0x )
= ( c_2Eseq_2Esuminf
@ ^ [V1n: tyop_2Enum_2Enum] :
( c_2Erealax_2Ereal__mul
@ ( ^ [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V2n ) ) )
@ V1n )
@ ( c_2Ereal_2Epow @ V0x @ V1n ) ) ) ) ).
thf(thm_2Etransc_2Ecos,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Ecos @ V0x )
= ( c_2Eseq_2Esuminf
@ ^ [V1n: tyop_2Enum_2Enum] :
( c_2Erealax_2Ereal__mul
@ ( ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2EEVEN @ V2n ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2EDIV @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V2n ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
@ V1n )
@ ( c_2Ereal_2Epow @ V0x @ V1n ) ) ) ) ).
thf(thm_2Etransc_2Esin,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Esin @ V0x )
= ( c_2Eseq_2Esuminf
@ ^ [V1n: tyop_2Enum_2Enum] :
( c_2Erealax_2Ereal__mul
@ ( ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2EEVEN @ V2n ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V2n ) ) ) )
@ V1n )
@ ( c_2Ereal_2Epow @ V0x @ V1n ) ) ) ) ).
thf(thm_2Etransc_2Eln,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eln @ V0x )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V1u: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Eexp @ V1u ) @ V0x ) ) ) ).
thf(thm_2Etransc_2Edivision,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ V2D )
<=> ( ( ( V2D @ c_2Enum_2E0 )
= V0a )
& ? [V3N: tyop_2Enum_2Enum] :
( ! [V4n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V4n @ V3N )
=> ( c_2Erealax_2Ereal__lt @ ( V2D @ V4n ) @ ( V2D @ ( c_2Enum_2ESUC @ V4n ) ) ) )
& ! [V5n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V5n @ V3N )
=> ( ( V2D @ V5n )
= V1b ) ) ) ) ) ).
thf(thm_2Etransc_2Edsize,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Edsize @ V0D )
= ( c_2Emin_2E_40 @ tyop_2Enum_2Enum
@ ^ [V1N: tyop_2Enum_2Enum] :
( c_2Ebool_2E_2F_5C
@ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Eprim__rec_2E_3C @ V2n @ V1N ) @ ( c_2Erealax_2Ereal__lt @ ( V0D @ V2n ) @ ( V0D @ ( c_2Enum_2ESUC @ V2n ) ) ) ) )
@ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Earithmetic_2E_3E_3D @ V3n @ V1N ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0D @ V3n ) @ ( V0D @ V1N ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2Etdiv,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3p: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Etdiv @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V2D @ V3p ) )
<=> ( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ V2D )
& ! [V4n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte @ ( V2D @ V4n ) @ ( V3p @ V4n ) )
& ( c_2Ereal_2Ereal__lte @ ( V3p @ V4n ) @ ( V2D @ ( c_2Enum_2ESUC @ V4n ) ) ) ) ) ) ).
thf(thm_2Etransc_2Egauge,axiom,
! [V0E: tyop_2Erealax_2Ereal > $o,V1g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Egauge @ V0E @ V1g )
<=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( V0E @ V2x )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V1g @ V2x ) ) ) ) ).
thf(thm_2Etransc_2Efine,axiom,
! [V0g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2p: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Efine @ V0g @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V1D @ V2p ) )
<=> ! [V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3n @ ( c_2Etransc_2Edsize @ V1D ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__sub @ ( V1D @ ( c_2Enum_2ESUC @ V3n ) ) @ ( V1D @ V3n ) ) @ ( V0g @ ( V2p @ V3n ) ) ) ) ) ).
thf(thm_2Etransc_2Ersum,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1p: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Ersum @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V0D @ V1p ) @ V2f )
= ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Etransc_2Edsize @ V0D ) )
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V2f @ ( V1p @ V3n ) ) @ ( c_2Ereal_2Ereal__sub @ ( V0D @ ( c_2Enum_2ESUC @ V3n ) ) @ ( V0D @ V3n ) ) ) ) ) ).
thf(thm_2Etransc_2EDint,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3k: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2EDint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ V2f @ V3k )
<=> ! [V4e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4e )
=> ? [V5g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Egauge
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V0a @ V6x ) @ ( c_2Ereal_2Ereal__lte @ V6x @ V1b ) )
@ V5g )
& ! [V7D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V8p: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Etransc_2Etdiv @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V7D @ V8p ) )
& ( c_2Etransc_2Efine @ V5g @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V7D @ V8p ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Etransc_2Ersum @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V7D @ V8p ) @ V2f ) @ V3k ) ) @ V4e ) ) ) ) ) ).
thf(thm_2Etransc_2Erpow__def,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Erpow @ V0a @ V1b )
= ( c_2Etransc_2Eexp @ ( c_2Erealax_2Ereal__mul @ V1b @ ( c_2Etransc_2Eln @ V0a ) ) ) ) ).
thf(thm_2Etransc_2ESIN__COS__SQRT,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Esin @ V0x ) )
=> ( ( c_2Etransc_2Esin @ V0x )
= ( c_2Etransc_2Esqrt @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2ECOS__SIN__SQRT,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Ecos @ V0x ) )
=> ( ( c_2Etransc_2Ecos @ V0x )
= ( c_2Etransc_2Esqrt @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Etransc_2Esin @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2ESIN__ACS__NZ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( (~)
@ ( ( c_2Etransc_2Esin @ ( c_2Etransc_2Eacs @ V0x ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Etransc_2ECOS__ASN__NZ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( (~)
@ ( ( c_2Etransc_2Ecos @ ( c_2Etransc_2Easn @ V0x ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Etransc_2ECOS__ATN__NZ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( (~)
@ ( ( c_2Etransc_2Ecos @ ( c_2Etransc_2Eatn @ V0x ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Etransc_2ECOS__SIN__SQ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( ( c_2Etransc_2Ecos @ V0x )
= ( c_2Etransc_2Esqrt @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Etransc_2Esin @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2ESIN__COS__SQ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ c_2Etransc_2Epi ) )
=> ( ( c_2Etransc_2Esin @ V0x )
= ( c_2Etransc_2Esqrt @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2ETAN__SEC,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Etransc_2Ecos @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Etransc_2Etan @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Epow @ ( c_2Erealax_2Einv @ ( c_2Etransc_2Ecos @ V0x ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Etransc_2ETAN__ATN,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( ( c_2Etransc_2Eatn @ ( c_2Etransc_2Etan @ V0x ) )
= V0x ) ) ).
thf(thm_2Etransc_2EATN__BOUNDS,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Etransc_2Eatn @ V0y ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Etransc_2Eatn @ V0y ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Etransc_2EATN__TAN,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Etan @ ( c_2Etransc_2Eatn @ V0y ) )
= V0y ) ).
thf(thm_2Etransc_2EATN,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Etransc_2Eatn @ V0y ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Etransc_2Eatn @ V0y ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( c_2Etransc_2Etan @ ( c_2Etransc_2Eatn @ V0y ) )
= V0y ) ) ).
thf(thm_2Etransc_2ECOS__ACS,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ c_2Etransc_2Epi ) )
=> ( ( c_2Etransc_2Eacs @ ( c_2Etransc_2Ecos @ V0x ) )
= V0x ) ) ).
thf(thm_2Etransc_2EACS__BOUNDS__LT,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0y )
& ( c_2Erealax_2Ereal__lt @ V0y @ ( 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_2Etransc_2Eacs @ V0y ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Etransc_2Eacs @ V0y ) @ c_2Etransc_2Epi ) ) ) ).
thf(thm_2Etransc_2EACS__BOUNDS,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0y )
& ( c_2Ereal_2Ereal__lte @ V0y @ ( 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_2Etransc_2Eacs @ V0y ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Eacs @ V0y ) @ c_2Etransc_2Epi ) ) ) ).
thf(thm_2Etransc_2EACS__COS,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0y )
& ( c_2Ereal_2Ereal__lte @ V0y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( c_2Etransc_2Ecos @ ( c_2Etransc_2Eacs @ V0y ) )
= V0y ) ) ).
thf(thm_2Etransc_2EACS,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0y )
& ( c_2Ereal_2Ereal__lte @ V0y @ ( 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_2Etransc_2Eacs @ V0y ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Eacs @ V0y ) @ c_2Etransc_2Epi )
& ( ( c_2Etransc_2Ecos @ ( c_2Etransc_2Eacs @ V0y ) )
= V0y ) ) ) ).
thf(thm_2Etransc_2ESIN__ASN,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( ( c_2Etransc_2Easn @ ( c_2Etransc_2Esin @ V0x ) )
= V0x ) ) ).
thf(thm_2Etransc_2EASN__BOUNDS__LT,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0y )
& ( c_2Erealax_2Ereal__lt @ V0y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Etransc_2Easn @ V0y ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Etransc_2Easn @ V0y ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2EASN__BOUNDS,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0y )
& ( c_2Ereal_2Ereal__lte @ V0y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Etransc_2Easn @ V0y ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Easn @ V0y ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2EASN__SIN,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0y )
& ( c_2Ereal_2Ereal__lte @ V0y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( c_2Etransc_2Esin @ ( c_2Etransc_2Easn @ V0y ) )
= V0y ) ) ).
thf(thm_2Etransc_2EASN,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0y )
& ( c_2Ereal_2Ereal__lte @ V0y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Etransc_2Easn @ V0y ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Easn @ V0y ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( c_2Etransc_2Esin @ ( c_2Etransc_2Easn @ V0y ) )
= V0y ) ) ) ).
thf(thm_2Etransc_2ETAN__TOTAL,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( c_2Ebool_2E_3F_21 @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ V1x ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ V1x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Etan @ V1x ) @ V0y ) ) ) ) ).
thf(thm_2Etransc_2ETAN__TOTAL__POS,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( 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 )
& ( c_2Erealax_2Ereal__lt @ V1x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( c_2Etransc_2Etan @ V1x )
= V0y ) ) ) ).
thf(thm_2Etransc_2ETAN__TOTAL__LEMMA,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0y )
=> ? [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
& ( c_2Erealax_2Ereal__lt @ V1x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( c_2Erealax_2Ereal__lt @ V0y @ ( c_2Etransc_2Etan @ V1x ) ) ) ) ).
thf(thm_2Etransc_2EDIFF__TAN,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Etransc_2Ecos @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( c_2Elim_2Ediffl @ c_2Etransc_2Etan @ ( c_2Erealax_2Einv @ ( c_2Ereal_2Epow @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0x ) ) ).
thf(thm_2Etransc_2ETAN__POS__PI2,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_2E_2F @ c_2Etransc_2Epi @ ( 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 ) @ ( c_2Etransc_2Etan @ V0x ) ) ) ).
thf(thm_2Etransc_2ETAN__DOUBLE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( ( c_2Etransc_2Ecos @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( ( c_2Etransc_2Ecos @ ( 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_2Enum_2E0 ) ) ) )
=> ( ( c_2Etransc_2Etan @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) )
= ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Etransc_2Etan @ V0x ) ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Etransc_2Etan @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2ETAN__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( ( c_2Etransc_2Ecos @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( ( c_2Etransc_2Ecos @ V1y )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Etransc_2Etan @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__add @ ( c_2Etransc_2Etan @ V0x ) @ ( c_2Etransc_2Etan @ V1y ) ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Etan @ V0x ) @ ( c_2Etransc_2Etan @ V1y ) ) ) ) ) ) ).
thf(thm_2Etransc_2ETAN__PERIODIC,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Etan @ ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ c_2Etransc_2Epi ) ) )
= ( c_2Etransc_2Etan @ V0x ) ) ).
thf(thm_2Etransc_2ETAN__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Etan @ ( c_2Erealax_2Ereal__neg @ V0x ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Etransc_2Etan @ V0x ) ) ) ).
thf(thm_2Etransc_2ETAN__NPI,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Etransc_2Etan @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ c_2Etransc_2Epi ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2ETAN__PI,axiom,
( ( c_2Etransc_2Etan @ c_2Etransc_2Epi )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2ETAN__0,axiom,
( ( c_2Etransc_2Etan @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2ESIN__ZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Etransc_2Esin @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ? [V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ V1n )
& ( V0x
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
| ? [V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ V2n )
& ( V0x
= ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2n ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2ECOS__ZERO,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Etransc_2Ecos @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( ? [V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Earithmetic_2EEVEN @ V1n ) )
& ( V0x
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
| ? [V2n: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Earithmetic_2EEVEN @ V2n ) )
& ( V0x
= ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2n ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2ESIN__ZERO__LEMMA,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( ( c_2Etransc_2Esin @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ? [V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ V1n )
& ( V0x
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2ECOS__ZERO__LEMMA,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( ( c_2Etransc_2Ecos @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ? [V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( c_2Earithmetic_2EEVEN @ V1n ) )
& ( V0x
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2ESIN__TOTAL,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0y )
& ( c_2Ereal_2Ereal__lte @ V0y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Ebool_2E_3F_21 @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ V1x ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V1x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Esin @ V1x ) @ V0y ) ) ) ) ) ).
thf(thm_2Etransc_2ECOS__TOTAL,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0y )
& ( c_2Ereal_2Ereal__lte @ V0y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Ebool_2E_3F_21 @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V1x @ c_2Etransc_2Epi ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V1x ) @ V0y ) ) ) ) ) ).
thf(thm_2Etransc_2ESIN__POS__PI__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ c_2Etransc_2Epi ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Esin @ V0x ) ) ) ).
thf(thm_2Etransc_2ESIN__POS__PI2__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Esin @ V0x ) ) ) ).
thf(thm_2Etransc_2ECOS__POS__PI__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Ecos @ V0x ) ) ) ).
thf(thm_2Etransc_2ECOS__POS__PI2__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Ecos @ V0x ) ) ) ).
thf(thm_2Etransc_2ESIN__POS__PI,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ c_2Etransc_2Epi ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Esin @ V0x ) ) ) ).
thf(thm_2Etransc_2ECOS__POS__PI,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( 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 ) @ ( c_2Etransc_2Ecos @ V0x ) ) ) ).
thf(thm_2Etransc_2ECOS__POS__PI2,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_2E_2F @ c_2Etransc_2Epi @ ( 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 ) @ ( c_2Etransc_2Ecos @ V0x ) ) ) ).
thf(thm_2Etransc_2ESIN__POS__PI2,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_2E_2F @ c_2Etransc_2Epi @ ( 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 ) @ ( c_2Etransc_2Esin @ V0x ) ) ) ).
thf(thm_2Etransc_2ESIN__NPI,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Etransc_2Esin @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ c_2Etransc_2Epi ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2ECOS__NPI,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ c_2Etransc_2Epi ) )
= ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0n ) ) ).
thf(thm_2Etransc_2ECOS__PERIODIC,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ c_2Etransc_2Epi ) ) )
= ( c_2Etransc_2Ecos @ V0x ) ) ).
thf(thm_2Etransc_2ESIN__PERIODIC,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Esin @ ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ c_2Etransc_2Epi ) ) )
= ( c_2Etransc_2Esin @ V0x ) ) ).
thf(thm_2Etransc_2ECOS__PERIODIC__PI,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__add @ V0x @ c_2Etransc_2Epi ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Etransc_2Ecos @ V0x ) ) ) ).
thf(thm_2Etransc_2ESIN__PERIODIC__PI,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Esin @ ( c_2Erealax_2Ereal__add @ V0x @ c_2Etransc_2Epi ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Etransc_2Esin @ V0x ) ) ) ).
thf(thm_2Etransc_2ECOS__SIN,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Ecos @ V0x )
= ( c_2Etransc_2Esin @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0x ) ) ) ).
thf(thm_2Etransc_2ESIN__COS,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Esin @ V0x )
= ( c_2Etransc_2Ecos @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0x ) ) ) ).
thf(thm_2Etransc_2ESIN__PI,axiom,
( ( c_2Etransc_2Esin @ c_2Etransc_2Epi )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2ECOS__PI,axiom,
( ( c_2Etransc_2Ecos @ c_2Etransc_2Epi )
= ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Etransc_2ESIN__PI2,axiom,
( ( c_2Etransc_2Esin @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( 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_2Etransc_2EPI__POS,axiom,
c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ c_2Etransc_2Epi ).
thf(thm_2Etransc_2EPI2__BOUNDS,axiom,
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( 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 ) ) ) ) ) ).
thf(thm_2Etransc_2ECOS__PI2,axiom,
( ( c_2Etransc_2Ecos @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2EPI2,axiom,
( ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ) ).
thf(thm_2Etransc_2ECOS__ISZERO,axiom,
( c_2Ebool_2E_3F_21 @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Etransc_2ECOS__2,axiom,
c_2Erealax_2Ereal__lt @ ( c_2Etransc_2Ecos @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ).
thf(thm_2Etransc_2ECOS__PAIRED,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( c_2Eseq_2Esums
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V1n ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1n ) ) ) ) @ ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1n ) ) )
@ ( c_2Etransc_2Ecos @ V0x ) ) ).
thf(thm_2Etransc_2ESIN__POS,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_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Esin @ V0x ) ) ) ).
thf(thm_2Etransc_2ESIN__PAIRED,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( c_2Eseq_2Esums
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V1n ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
@ ( c_2Etransc_2Esin @ V0x ) ) ).
thf(thm_2Etransc_2ECOS__DOUBLE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Epow @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Etransc_2Esin @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Etransc_2ESIN__DOUBLE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Esin @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Esin @ V0x ) @ ( c_2Etransc_2Ecos @ V0x ) ) ) ) ).
thf(thm_2Etransc_2ECOS__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__neg @ V0x ) )
= ( c_2Etransc_2Ecos @ V0x ) ) ).
thf(thm_2Etransc_2ESIN__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Esin @ ( c_2Erealax_2Ereal__neg @ V0x ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Etransc_2Esin @ V0x ) ) ) ).
thf(thm_2Etransc_2ECOS__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Etransc_2Ecos @ V1y ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Esin @ V0x ) @ ( c_2Etransc_2Esin @ V1y ) ) ) ) ).
thf(thm_2Etransc_2ESIN__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Esin @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Esin @ V0x ) @ ( c_2Etransc_2Ecos @ V1y ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Etransc_2Esin @ V1y ) ) ) ) ).
thf(thm_2Etransc_2ESIN__COS__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__add @ ( c_2Etransc_2Esin @ ( c_2Erealax_2Ereal__neg @ V0x ) ) @ ( c_2Etransc_2Esin @ V0x ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__sub @ ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__neg @ V0x ) ) @ ( c_2Etransc_2Ecos @ V0x ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2ESIN__COS__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__sub @ ( c_2Etransc_2Esin @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Esin @ V0x ) @ ( c_2Etransc_2Ecos @ V1y ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Etransc_2Esin @ V1y ) ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__sub @ ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Etransc_2Ecos @ V1y ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Esin @ V0x ) @ ( c_2Etransc_2Esin @ V1y ) ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2ECOS__BOUNDS,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Etransc_2Ecos @ V0x ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Etransc_2ECOS__BOUND,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Etransc_2Ecos @ V0x ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Etransc_2ESIN__BOUNDS,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Etransc_2Esin @ V0x ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Esin @ V0x ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Etransc_2ESIN__BOUND,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Etransc_2Esin @ V0x ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Etransc_2ESIN__CIRCLE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ ( c_2Etransc_2Esin @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Etransc_2Ecos @ V0x ) @ ( 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_2Etransc_2ECOS__0,axiom,
( ( c_2Etransc_2Ecos @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Etransc_2ESIN__0,axiom,
( ( c_2Etransc_2Esin @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2ESQRT__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= V1y )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) )
=> ( V0x
= ( c_2Etransc_2Esqrt @ V1y ) ) ) ).
thf(thm_2Etransc_2EREAL__DIV__SQRT,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( ( c_2Ereal_2E_2F @ V0x @ ( c_2Etransc_2Esqrt @ V0x ) )
= ( c_2Etransc_2Esqrt @ V0x ) ) ) ).
thf(thm_2Etransc_2ESQRT__EVEN__POW2,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2EEVEN @ V0n )
=> ( ( c_2Etransc_2Esqrt @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0n ) )
= ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2EDIV @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Etransc_2ESQRT__MONO__LE,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 @ V0x @ V1y ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Esqrt @ V0x ) @ ( c_2Etransc_2Esqrt @ V1y ) ) ) ).
thf(thm_2Etransc_2ESQRT__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_2Etransc_2Esqrt @ ( c_2Ereal_2E_2F @ V0x @ V1y ) )
= ( c_2Ereal_2E_2F @ ( c_2Etransc_2Esqrt @ V0x ) @ ( c_2Etransc_2Esqrt @ V1y ) ) ) ) ).
thf(thm_2Etransc_2ESQRT__INV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( ( c_2Etransc_2Esqrt @ ( c_2Erealax_2Einv @ V0x ) )
= ( c_2Erealax_2Einv @ ( c_2Etransc_2Esqrt @ V0x ) ) ) ) ).
thf(thm_2Etransc_2ESQRT__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_2Etransc_2Esqrt @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Esqrt @ V0x ) @ ( c_2Etransc_2Esqrt @ V1y ) ) ) ) ).
thf(thm_2Etransc_2ESQRT__POS__UNIQ,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_2Epow @ V1y @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= V0x ) )
=> ( ( c_2Etransc_2Esqrt @ V0x )
= V1y ) ) ).
thf(thm_2Etransc_2EPOW__2__SQRT,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( ( c_2Etransc_2Esqrt @ ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= V0x ) ) ).
thf(thm_2Etransc_2ESQRT__POW__2,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( ( c_2Ereal_2Epow @ ( c_2Etransc_2Esqrt @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= V0x ) ) ).
thf(thm_2Etransc_2ESQRT__POW2,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Epow @ ( c_2Etransc_2Esqrt @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= V0x )
<=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Etransc_2ESQRT__POS__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_2Enum_2E0 ) @ ( c_2Etransc_2Esqrt @ V0x ) ) ) ).
thf(thm_2Etransc_2ESQRT__POS__LT,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_2Etransc_2Esqrt @ V0x ) ) ) ).
thf(thm_2Etransc_2ESQRT__1,axiom,
( ( c_2Etransc_2Esqrt @ ( 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_2Etransc_2ESQRT__0,axiom,
( ( c_2Etransc_2Esqrt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2EROOT__MONO__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_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ V1x ) @ ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ V2y ) ) ) ).
thf(thm_2Etransc_2EROOT__DIV,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_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Ereal_2E_2F @ V1x @ V2y ) )
= ( c_2Ereal_2E_2F @ ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ V1x ) @ ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ V2y ) ) ) ) ).
thf(thm_2Etransc_2EROOT__INV,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
=> ( ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Erealax_2Einv @ V1x ) )
= ( c_2Erealax_2Einv @ ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ V1x ) ) ) ) ).
thf(thm_2Etransc_2EROOT__MUL,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_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Erealax_2Ereal__mul @ V1x @ V2y ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ V1x ) @ ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ V2y ) ) ) ) ).
thf(thm_2Etransc_2EROOT__POS__UNIQ,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 @ V2y @ ( c_2Enum_2ESUC @ V0n ) )
= V1x ) )
=> ( ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ V1x )
= V2y ) ) ).
thf(thm_2Etransc_2EROOT__POS,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: 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 ) @ ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ V1x ) ) ) ).
thf(thm_2Etransc_2EPOW__ROOT__POS,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
=> ( ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Ereal_2Epow @ V1x @ ( c_2Enum_2ESUC @ V0n ) ) )
= V1x ) ) ).
thf(thm_2Etransc_2EROOT__POW__POS,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
=> ( ( c_2Ereal_2Epow @ ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ V1x ) @ ( c_2Enum_2ESUC @ V0n ) )
= V1x ) ) ).
thf(thm_2Etransc_2EROOT__POS__LT,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ V1x ) ) ) ).
thf(thm_2Etransc_2EROOT__1,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ ( 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_2Etransc_2EROOT__0,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2EROOT__LN,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
=> ( ( c_2Etransc_2Eroot @ ( c_2Enum_2ESUC @ V0n ) @ V1x )
= ( c_2Etransc_2Eexp @ ( c_2Ereal_2E_2F @ ( c_2Etransc_2Eln @ V1x ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Enum_2ESUC @ V0n ) ) ) ) ) ) ).
thf(thm_2Etransc_2EROOT__LT__LEMMA,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
=> ( ( c_2Ereal_2Epow @ ( c_2Etransc_2Eexp @ ( c_2Ereal_2E_2F @ ( c_2Etransc_2Eln @ V1x ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Enum_2ESUC @ V0n ) ) ) ) @ ( c_2Enum_2ESUC @ V0n ) )
= V1x ) ) ).
thf(thm_2Etransc_2EEXP__CONVERGES,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( c_2Eseq_2Esums
@ ^ [V1n: tyop_2Enum_2Enum] :
( c_2Erealax_2Ereal__mul
@ ( ^ [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V2n ) ) )
@ V1n )
@ ( c_2Ereal_2Epow @ V0x @ V1n ) )
@ ( c_2Etransc_2Eexp @ V0x ) ) ).
thf(thm_2Etransc_2ESIN__CONVERGES,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( c_2Eseq_2Esums
@ ^ [V1n: tyop_2Enum_2Enum] :
( c_2Erealax_2Ereal__mul
@ ( ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2EEVEN @ V2n ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V2n ) ) ) )
@ V1n )
@ ( c_2Ereal_2Epow @ V0x @ V1n ) )
@ ( c_2Etransc_2Esin @ V0x ) ) ).
thf(thm_2Etransc_2ECOS__CONVERGES,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( c_2Eseq_2Esums
@ ^ [V1n: tyop_2Enum_2Enum] :
( c_2Erealax_2Ereal__mul
@ ( ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2EEVEN @ V2n ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2EDIV @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V2n ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
@ V1n )
@ ( c_2Ereal_2Epow @ V0x @ V1n ) )
@ ( c_2Etransc_2Ecos @ V0x ) ) ).
thf(thm_2Etransc_2EEXP__FDIFF,axiom,
( ( c_2Epowser_2Ediffs
@ ^ [V0n: tyop_2Enum_2Enum] : ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V0n ) ) ) )
= ( ^ [V1n: tyop_2Enum_2Enum] : ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V1n ) ) ) ) ) ).
thf(thm_2Etransc_2ESIN__FDIFF,axiom,
( ( c_2Epowser_2Ediffs
@ ^ [V0n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2EEVEN @ V0n ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V0n ) ) ) ) )
= ( ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2EEVEN @ V1n ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2EDIV @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V1n ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Etransc_2ECOS__FDIFF,axiom,
( ( c_2Epowser_2Ediffs
@ ^ [V0n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2EEVEN @ V0n ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2EDIV @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V0n ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
= ( ^ [V1n: tyop_2Enum_2Enum] :
( c_2Erealax_2Ereal__neg
@ ( ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2EEVEN @ V2n ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V2n ) ) ) )
@ V1n ) ) ) ) ).
thf(thm_2Etransc_2ESIN__NEGLEMMA,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg @ ( c_2Etransc_2Esin @ V0x ) )
= ( c_2Eseq_2Esuminf
@ ^ [V1n: tyop_2Enum_2Enum] :
( c_2Erealax_2Ereal__neg
@ ( c_2Erealax_2Ereal__mul
@ ( ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2EEVEN @ V2n ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V2n ) ) ) )
@ V1n )
@ ( c_2Ereal_2Epow @ V0x @ V1n ) ) ) ) ) ).
thf(thm_2Etransc_2EDIFF__EXP,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Elim_2Ediffl @ c_2Etransc_2Eexp @ ( c_2Etransc_2Eexp @ V0x ) @ V0x ) ).
thf(thm_2Etransc_2EDIFF__SIN,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Elim_2Ediffl @ c_2Etransc_2Esin @ ( c_2Etransc_2Ecos @ V0x ) @ V0x ) ).
thf(thm_2Etransc_2EDIFF__COS,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Elim_2Ediffl @ c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__neg @ ( c_2Etransc_2Esin @ V0x ) ) @ V0x ) ).
thf(thm_2Etransc_2EDIFF__COMPOSITE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum,V2m: tyop_2Erealax_2Ereal,V3l: tyop_2Erealax_2Ereal,V4g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V5f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( ( c_2Elim_2Ediffl @ V5f @ V3l @ V0x )
& ( (~)
@ ( ( V5f @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Elim_2Ediffl
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Einv @ ( V5f @ V6x ) )
@ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ V3l @ ( c_2Ereal_2Epow @ ( V5f @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) )
@ V0x ) )
& ( ( ( c_2Elim_2Ediffl @ V5f @ V3l @ V0x )
& ( c_2Elim_2Ediffl @ V4g @ V2m @ V0x )
& ( (~)
@ ( ( V4g @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Elim_2Ediffl
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F @ ( V5f @ V7x ) @ ( V4g @ V7x ) )
@ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ V3l @ ( V4g @ V0x ) ) @ ( c_2Erealax_2Ereal__mul @ V2m @ ( V5f @ V0x ) ) ) @ ( c_2Ereal_2Epow @ ( V4g @ V0x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
@ V0x ) )
& ( ( ( c_2Elim_2Ediffl @ V5f @ V3l @ V0x )
& ( c_2Elim_2Ediffl @ V4g @ V2m @ V0x ) )
=> ( c_2Elim_2Ediffl
@ ^ [V8x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ ( V5f @ V8x ) @ ( V4g @ V8x ) )
@ ( c_2Erealax_2Ereal__add @ V3l @ V2m )
@ V0x ) )
& ( ( ( c_2Elim_2Ediffl @ V5f @ V3l @ V0x )
& ( c_2Elim_2Ediffl @ V4g @ V2m @ V0x ) )
=> ( c_2Elim_2Ediffl
@ ^ [V9x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ ( V5f @ V9x ) @ ( V4g @ V9x ) )
@ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V3l @ ( V4g @ V0x ) ) @ ( c_2Erealax_2Ereal__mul @ V2m @ ( V5f @ V0x ) ) )
@ V0x ) )
& ( ( ( c_2Elim_2Ediffl @ V5f @ V3l @ V0x )
& ( c_2Elim_2Ediffl @ V4g @ V2m @ V0x ) )
=> ( c_2Elim_2Ediffl
@ ^ [V10x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub @ ( V5f @ V10x ) @ ( V4g @ V10x ) )
@ ( c_2Ereal_2Ereal__sub @ V3l @ V2m )
@ V0x ) )
& ( ( c_2Elim_2Ediffl @ V5f @ V3l @ V0x )
=> ( c_2Elim_2Ediffl
@ ^ [V11x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ ( V5f @ V11x ) )
@ ( c_2Erealax_2Ereal__neg @ V3l )
@ V0x ) )
& ( ( c_2Elim_2Ediffl @ V4g @ V2m @ V0x )
=> ( c_2Elim_2Ediffl
@ ^ [V12x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Epow @ ( V4g @ V12x ) @ V1n )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ ( c_2Ereal_2Epow @ ( V4g @ V0x ) @ ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ V2m )
@ V0x ) )
& ( ( c_2Elim_2Ediffl @ V4g @ V2m @ V0x )
=> ( c_2Elim_2Ediffl
@ ^ [V13x: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eexp @ ( V4g @ V13x ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Eexp @ ( V4g @ V0x ) ) @ V2m )
@ V0x ) )
& ( ( c_2Elim_2Ediffl @ V4g @ V2m @ V0x )
=> ( c_2Elim_2Ediffl
@ ^ [V14x: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Esin @ ( V4g @ V14x ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Ecos @ ( V4g @ V0x ) ) @ V2m )
@ V0x ) )
& ( ( c_2Elim_2Ediffl @ V4g @ V2m @ V0x )
=> ( c_2Elim_2Ediffl
@ ^ [V15x: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Ecos @ ( V4g @ V15x ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__neg @ ( c_2Etransc_2Esin @ ( V4g @ V0x ) ) ) @ V2m )
@ V0x ) ) ) ).
thf(thm_2Etransc_2EEXP__0,axiom,
( ( c_2Etransc_2Eexp @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Etransc_2EEXP__LE__X,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) @ ( c_2Etransc_2Eexp @ V0x ) ) ) ).
thf(thm_2Etransc_2EEXP__LT__1,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_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Etransc_2Eexp @ V0x ) ) ) ).
thf(thm_2Etransc_2EEXP__ADD__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Eexp @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) @ ( c_2Etransc_2Eexp @ ( c_2Erealax_2Ereal__neg @ V0x ) ) )
= ( c_2Etransc_2Eexp @ V1y ) ) ).
thf(thm_2Etransc_2EEXP__NEG__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Eexp @ V0x ) @ ( c_2Etransc_2Eexp @ ( c_2Erealax_2Ereal__neg @ V0x ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Etransc_2EEXP__NEG__MUL2,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Eexp @ ( c_2Erealax_2Ereal__neg @ V0x ) ) @ ( c_2Etransc_2Eexp @ V0x ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Etransc_2EEXP__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eexp @ ( c_2Erealax_2Ereal__neg @ V0x ) )
= ( c_2Erealax_2Einv @ ( c_2Etransc_2Eexp @ V0x ) ) ) ).
thf(thm_2Etransc_2EEXP__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eexp @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Eexp @ V0x ) @ ( c_2Etransc_2Eexp @ V1y ) ) ) ).
thf(thm_2Etransc_2EEXP__POS__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Eexp @ V0x ) ) ).
thf(thm_2Etransc_2EEXP__NZ,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( (~)
@ ( ( c_2Etransc_2Eexp @ V0x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Etransc_2EEXP__POS__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Eexp @ V0x ) ) ).
thf(thm_2Etransc_2EEXP__N,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eexp @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ V1x ) )
= ( c_2Ereal_2Epow @ ( c_2Etransc_2Eexp @ V1x ) @ V0n ) ) ).
thf(thm_2Etransc_2EEXP__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eexp @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) )
= ( c_2Ereal_2E_2F @ ( c_2Etransc_2Eexp @ V0x ) @ ( c_2Etransc_2Eexp @ V1y ) ) ) ).
thf(thm_2Etransc_2EEXP__MONO__IMP,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Etransc_2Eexp @ V0x ) @ ( c_2Etransc_2Eexp @ V1y ) ) ) ).
thf(thm_2Etransc_2EEXP__MONO__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Etransc_2Eexp @ V0x ) @ ( c_2Etransc_2Eexp @ V1y ) )
= ( c_2Erealax_2Ereal__lt @ V0x @ V1y ) ) ).
thf(thm_2Etransc_2EEXP__MONO__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Eexp @ V0x ) @ ( c_2Etransc_2Eexp @ V1y ) )
= ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ).
thf(thm_2Etransc_2EEXP__INJ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Etransc_2Eexp @ V0x )
= ( c_2Etransc_2Eexp @ V1y ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Etransc_2EEXP__TOTAL__LEMMA,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0y )
=> ? [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
& ( c_2Ereal_2Ereal__lte @ V1x @ ( c_2Ereal_2Ereal__sub @ V0y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ( ( c_2Etransc_2Eexp @ V1x )
= V0y ) ) ) ).
thf(thm_2Etransc_2EEXP__TOTAL,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0y )
=> ? [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eexp @ V1x )
= V0y ) ) ).
thf(thm_2Etransc_2ELN__EXP,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eln @ ( c_2Etransc_2Eexp @ V0x ) )
= V0x ) ).
thf(thm_2Etransc_2EEXP__LN,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Etransc_2Eexp @ ( c_2Etransc_2Eln @ V0x ) )
= V0x )
<=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x ) ) ).
thf(thm_2Etransc_2ELN__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_2Etransc_2Eln @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Etransc_2Eln @ V0x ) @ ( c_2Etransc_2Eln @ V1y ) ) ) ) ).
thf(thm_2Etransc_2ELN__INJ,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_2Etransc_2Eln @ V0x )
= ( c_2Etransc_2Eln @ V1y ) )
<=> ( V0x = V1y ) ) ) ).
thf(thm_2Etransc_2ELN__1,axiom,
( ( c_2Etransc_2Eln @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2ELN__INV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( ( c_2Etransc_2Eln @ ( c_2Erealax_2Einv @ V0x ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Etransc_2Eln @ V0x ) ) ) ) ).
thf(thm_2Etransc_2ELN__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_2Etransc_2Eln @ ( c_2Ereal_2E_2F @ V0x @ V1y ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Etransc_2Eln @ V0x ) @ ( c_2Etransc_2Eln @ V1y ) ) ) ) ).
thf(thm_2Etransc_2ELN__MONO__LT,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_2Etransc_2Eln @ V0x ) @ ( c_2Etransc_2Eln @ V1y ) )
= ( c_2Erealax_2Ereal__lt @ V0x @ V1y ) ) ) ).
thf(thm_2Etransc_2ELN__MONO__LE,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_2Ereal_2Ereal__lte @ ( c_2Etransc_2Eln @ V0x ) @ ( c_2Etransc_2Eln @ V1y ) )
= ( c_2Ereal_2Ereal__lte @ V0x @ V1y ) ) ) ).
thf(thm_2Etransc_2ELN__POW,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1x )
=> ( ( c_2Etransc_2Eln @ ( c_2Ereal_2Epow @ V1x @ V0n ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Etransc_2Eln @ V1x ) ) ) ) ).
thf(thm_2Etransc_2ELN__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Eln @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x ) ) @ V0x ) ) ).
thf(thm_2Etransc_2ELN__LT__X,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Etransc_2Eln @ V0x ) @ V0x ) ) ).
thf(thm_2Etransc_2ELN__POS,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_2Enum_2E0 ) @ ( c_2Etransc_2Eln @ V0x ) ) ) ).
thf(thm_2Etransc_2EDIFF__LN,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( c_2Elim_2Ediffl @ c_2Etransc_2Eln @ ( c_2Erealax_2Einv @ V0x ) @ V0x ) ) ).
thf(thm_2Etransc_2EDIFF__ASN__LEMMA,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Elim_2Ediffl @ c_2Etransc_2Easn @ ( c_2Erealax_2Einv @ ( c_2Etransc_2Ecos @ ( c_2Etransc_2Easn @ V0x ) ) ) @ V0x ) ) ).
thf(thm_2Etransc_2EDIFF__ASN,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Elim_2Ediffl @ c_2Etransc_2Easn @ ( c_2Erealax_2Einv @ ( c_2Etransc_2Esqrt @ ( c_2Ereal_2Ereal__sub @ ( 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 ) ) ) ) ) ) @ V0x ) ) ).
thf(thm_2Etransc_2EDIFF__ACS__LEMMA,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Elim_2Ediffl @ c_2Etransc_2Eacs @ ( c_2Erealax_2Einv @ ( c_2Erealax_2Ereal__neg @ ( c_2Etransc_2Esin @ ( c_2Etransc_2Eacs @ V0x ) ) ) ) @ V0x ) ) ).
thf(thm_2Etransc_2EDIFF__ACS,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0x )
& ( c_2Erealax_2Ereal__lt @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( c_2Elim_2Ediffl @ c_2Etransc_2Eacs @ ( c_2Erealax_2Ereal__neg @ ( c_2Erealax_2Einv @ ( c_2Etransc_2Esqrt @ ( c_2Ereal_2Ereal__sub @ ( 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 ) ) ) ) ) ) ) @ V0x ) ) ).
thf(thm_2Etransc_2EDIFF__ATN,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Elim_2Ediffl @ c_2Etransc_2Eatn @ ( c_2Erealax_2Einv @ ( c_2Erealax_2Ereal__add @ ( 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 ) ) ) ) ) @ V0x ) ).
thf(thm_2Etransc_2EDIVISION__0,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( V0a = V1b )
=> ( ( c_2Etransc_2Edsize
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2n @ c_2Enum_2E0 ) @ V0a @ V1b ) )
= c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2EDIVISION__1,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0a @ V1b )
=> ( ( c_2Etransc_2Edsize
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2n @ c_2Enum_2E0 ) @ V0a @ V1b ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Etransc_2EDIVISION__SINGLE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ V0a @ V1b )
=> ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b )
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2n @ c_2Enum_2E0 ) @ V0a @ V1b ) ) ) ).
thf(thm_2Etransc_2EDIVISION__LHS,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ V0D )
=> ( ( V0D @ c_2Enum_2E0 )
= V1a ) ) ).
thf(thm_2Etransc_2EDIVISION__THM,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ V0D )
<=> ( ( ( V0D @ c_2Enum_2E0 )
= V1a )
& ! [V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3n @ ( c_2Etransc_2Edsize @ V0D ) )
=> ( c_2Erealax_2Ereal__lt @ ( V0D @ V3n ) @ ( V0D @ ( c_2Enum_2ESUC @ V3n ) ) ) )
& ! [V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V4n @ ( c_2Etransc_2Edsize @ V0D ) )
=> ( ( V0D @ V4n )
= V2b ) ) ) ) ).
thf(thm_2Etransc_2EDIVISION__RHS,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ V0D )
=> ( ( V0D @ ( c_2Etransc_2Edsize @ V0D ) )
= V2b ) ) ).
thf(thm_2Etransc_2EDIVISION__LT__GEN,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal,V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ V0D )
& ( c_2Eprim__rec_2E_3C @ V3m @ V4n )
& ( c_2Earithmetic_2E_3C_3D @ V4n @ ( c_2Etransc_2Edsize @ V0D ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( V0D @ V3m ) @ ( V0D @ V4n ) ) ) ).
thf(thm_2Etransc_2EDIVISION__LT,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ V0D )
=> ! [V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3n @ ( c_2Etransc_2Edsize @ V0D ) )
=> ( c_2Erealax_2Ereal__lt @ ( V0D @ c_2Enum_2E0 ) @ ( V0D @ ( c_2Enum_2ESUC @ V3n ) ) ) ) ) ).
thf(thm_2Etransc_2EDIVISION__LE,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ V0D )
=> ( c_2Ereal_2Ereal__lte @ V1a @ V2b ) ) ).
thf(thm_2Etransc_2EDIVISION__GT,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ V0D )
=> ! [V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3n @ ( c_2Etransc_2Edsize @ V0D ) )
=> ( c_2Erealax_2Ereal__lt @ ( V0D @ V3n ) @ ( V0D @ ( c_2Etransc_2Edsize @ V0D ) ) ) ) ) ).
thf(thm_2Etransc_2EDIVISION__EQ,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ V0D )
=> ( ( V1a = V2b )
<=> ( ( c_2Etransc_2Edsize @ V0D )
= c_2Enum_2E0 ) ) ) ).
thf(thm_2Etransc_2EDIVISION__LBOUND,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ V0D )
=> ! [V3r: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ V1a @ ( V0D @ V3r ) ) ) ).
thf(thm_2Etransc_2EDIVISION__LBOUND__LT,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ V0D )
& ( (~)
@ ( ( c_2Etransc_2Edsize @ V0D )
= c_2Enum_2E0 ) ) )
=> ! [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt @ V1a @ ( V0D @ ( c_2Enum_2ESUC @ V3n ) ) ) ) ).
thf(thm_2Etransc_2EDIVISION__UBOUND,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ V0D )
=> ! [V3r: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( V0D @ V3r ) @ V2b ) ) ).
thf(thm_2Etransc_2EDIVISION__UBOUND__LT,axiom,
! [V0D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal,V3n: tyop_2Enum_2Enum] :
( ( ( c_2Etransc_2Edivision @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ V0D )
& ( c_2Eprim__rec_2E_3C @ V3n @ ( c_2Etransc_2Edsize @ V0D ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( V0D @ V3n ) @ V2b ) ) ).
thf(thm_2Etransc_2EDIVISION__APPEND,axiom,
! [V0g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal,V3c: tyop_2Erealax_2Ereal] :
( ( ? [V4D1: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5p1: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Etdiv @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V2b ) @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V4D1 @ V5p1 ) )
& ( c_2Etransc_2Efine @ V0g @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V4D1 @ V5p1 ) ) )
& ? [V6D2: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V7p2: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Etdiv @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2b @ V3c ) @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V6D2 @ V7p2 ) )
& ( c_2Etransc_2Efine @ V0g @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V6D2 @ V7p2 ) ) ) )
=> ? [V8D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V9p: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Etdiv @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V3c ) @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V8D @ V9p ) )
& ( c_2Etransc_2Efine @ V0g @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V8D @ V9p ) ) ) ) ).
thf(thm_2Etransc_2EDIVISION__EXISTS,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0a @ V1b )
& ( c_2Etransc_2Egauge
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V0a @ V3x ) @ ( c_2Ereal_2Ereal__lte @ V3x @ V1b ) )
@ V2g ) )
=> ? [V4D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5p: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Etdiv @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V4D @ V5p ) )
& ( c_2Etransc_2Efine @ V2g @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V4D @ V5p ) ) ) ) ).
thf(thm_2Etransc_2EGAUGE__MIN,axiom,
! [V0E: tyop_2Erealax_2Ereal > $o,V1g1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2g2: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Etransc_2Egauge @ V0E @ V1g1 )
& ( c_2Etransc_2Egauge @ V0E @ V2g2 ) )
=> ( c_2Etransc_2Egauge @ V0E
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__lt @ ( V1g1 @ V3x ) @ ( V2g2 @ V3x ) ) @ ( V1g1 @ V3x ) @ ( V2g2 @ V3x ) ) ) ) ).
thf(thm_2Etransc_2EFINE__MIN,axiom,
! [V0g1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1g2: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2D: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V3p: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Efine
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__lt @ ( V0g1 @ V4x ) @ ( V1g2 @ V4x ) ) @ ( V0g1 @ V4x ) @ ( V1g2 @ V4x ) )
@ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V2D @ V3p ) )
=> ( ( c_2Etransc_2Efine @ V0g1 @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V2D @ V3p ) )
& ( c_2Etransc_2Efine @ V1g2 @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V2D @ V3p ) ) ) ) ).
thf(thm_2Etransc_2EDINT__UNIQ,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V3k1: tyop_2Erealax_2Ereal,V4k2: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V0a @ V1b )
& ( c_2Etransc_2EDint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ V2f @ V3k1 )
& ( c_2Etransc_2EDint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0a @ V1b ) @ V2f @ V4k2 ) )
=> ( V3k1 = V4k2 ) ) ).
thf(thm_2Etransc_2EINTEGRAL__NULL,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal] : ( c_2Etransc_2EDint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V1a @ V1a ) @ V0f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Etransc_2EFTC1,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1f_27: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2a: tyop_2Erealax_2Ereal,V3b: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V2a @ V3b )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Ereal__lte @ V2a @ V4x )
& ( c_2Ereal_2Ereal__lte @ V4x @ V3b ) )
=> ( c_2Elim_2Ediffl @ V0f @ ( V1f_27 @ V4x ) @ V4x ) ) )
=> ( c_2Etransc_2EDint @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V2a @ V3b ) @ V1f_27 @ ( c_2Ereal_2Ereal__sub @ ( V0f @ V3b ) @ ( V0f @ V2a ) ) ) ) ).
thf(thm_2Etransc_2EMCLAURIN,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1diff: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal,V3n: tyop_2Enum_2Enum] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2h )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V3n )
& ( ( V1diff @ c_2Enum_2E0 )
= V0f )
& ! [V4m: tyop_2Enum_2Enum,V5t: tyop_2Erealax_2Ereal] :
( ( ( c_2Eprim__rec_2E_3C @ V4m @ V3n )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5t )
& ( c_2Ereal_2Ereal__lte @ V5t @ V2h ) )
=> ( c_2Elim_2Ediffl @ ( V1diff @ V4m ) @ ( V1diff @ ( c_2Enum_2ESUC @ V4m ) @ V5t ) @ V5t ) ) )
=> ? [V6t: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V6t )
& ( c_2Erealax_2Ereal__lt @ V6t @ V2h )
& ( ( V0f @ V2h )
= ( c_2Erealax_2Ereal__add
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V3n )
@ ^ [V7m: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( V1diff @ V7m @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V7m ) ) ) @ ( c_2Ereal_2Epow @ V2h @ V7m ) ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( V1diff @ V3n @ V6t ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V3n ) ) ) @ ( c_2Ereal_2Epow @ V2h @ V3n ) ) ) ) ) ) ).
thf(thm_2Etransc_2EMCLAURIN__NEG,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1diff: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal,V3n: tyop_2Enum_2Enum] :
( ( ( c_2Erealax_2Ereal__lt @ V2h @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V3n )
& ( ( V1diff @ c_2Enum_2E0 )
= V0f )
& ! [V4m: tyop_2Enum_2Enum,V5t: tyop_2Erealax_2Ereal] :
( ( ( c_2Eprim__rec_2E_3C @ V4m @ V3n )
& ( c_2Ereal_2Ereal__lte @ V2h @ V5t )
& ( c_2Ereal_2Ereal__lte @ V5t @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( c_2Elim_2Ediffl @ ( V1diff @ V4m ) @ ( V1diff @ ( c_2Enum_2ESUC @ V4m ) @ V5t ) @ V5t ) ) )
=> ? [V6t: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V2h @ V6t )
& ( c_2Erealax_2Ereal__lt @ V6t @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( ( V0f @ V2h )
= ( c_2Erealax_2Ereal__add
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V3n )
@ ^ [V7m: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( V1diff @ V7m @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V7m ) ) ) @ ( c_2Ereal_2Epow @ V2h @ V7m ) ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( V1diff @ V3n @ V6t ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V3n ) ) ) @ ( c_2Ereal_2Epow @ V2h @ V3n ) ) ) ) ) ) ).
thf(thm_2Etransc_2EMCLAURIN__ALL__LT,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1diff: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( ( V1diff @ c_2Enum_2E0 )
= V0f )
& ! [V2m: tyop_2Enum_2Enum,V3x: tyop_2Erealax_2Ereal] : ( c_2Elim_2Ediffl @ ( V1diff @ V2m ) @ ( V1diff @ ( c_2Enum_2ESUC @ V2m ) @ V3x ) @ V3x ) )
=> ! [V4x: tyop_2Erealax_2Ereal,V5n: tyop_2Enum_2Enum] :
( ( ( (~)
@ ( V4x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V5n ) )
=> ? [V6t: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Eabs @ V6t ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V6t ) @ ( c_2Ereal_2Eabs @ V4x ) )
& ( ( V0f @ V4x )
= ( c_2Erealax_2Ereal__add
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V5n )
@ ^ [V7m: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( V1diff @ V7m @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V7m ) ) ) @ ( c_2Ereal_2Epow @ V4x @ V7m ) ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( V1diff @ V5n @ V6t ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V5n ) ) ) @ ( c_2Ereal_2Epow @ V4x @ V5n ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2EMCLAURIN__ZERO,axiom,
! [V0diff: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum,V2x: tyop_2Erealax_2Ereal] :
( ( ( V2x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n ) )
=> ( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1n )
@ ^ [V3m: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( V0diff @ V3m @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V3m ) ) ) @ ( c_2Ereal_2Epow @ V2x @ V3m ) ) )
= ( V0diff @ c_2Enum_2E0 @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Etransc_2EMCLAURIN__ALL__LE,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1diff: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( ( V1diff @ c_2Enum_2E0 )
= V0f )
& ! [V2m: tyop_2Enum_2Enum,V3x: tyop_2Erealax_2Ereal] : ( c_2Elim_2Ediffl @ ( V1diff @ V2m ) @ ( V1diff @ ( c_2Enum_2ESUC @ V2m ) @ V3x ) @ V3x ) )
=> ! [V4x: tyop_2Erealax_2Ereal,V5n: tyop_2Enum_2Enum] :
? [V6t: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V6t ) @ ( c_2Ereal_2Eabs @ V4x ) )
& ( ( V0f @ V4x )
= ( c_2Erealax_2Ereal__add
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V5n )
@ ^ [V7m: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( V1diff @ V7m @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V7m ) ) ) @ ( c_2Ereal_2Epow @ V4x @ V7m ) ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( V1diff @ V5n @ V6t ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V5n ) ) ) @ ( c_2Ereal_2Epow @ V4x @ V5n ) ) ) ) ) ) ).
thf(thm_2Etransc_2EMCLAURIN__EXP__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( ( (~)
@ ( V0x
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V1n ) )
=> ? [V2t: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Eabs @ V2t ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V2t ) @ ( c_2Ereal_2Eabs @ V0x ) )
& ( ( c_2Etransc_2Eexp @ V0x )
= ( c_2Erealax_2Ereal__add
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1n )
@ ^ [V3m: tyop_2Enum_2Enum] : ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ V0x @ V3m ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V3m ) ) ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( c_2Etransc_2Eexp @ V2t ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V1n ) ) ) @ ( c_2Ereal_2Epow @ V0x @ V1n ) ) ) ) ) ) ).
thf(thm_2Etransc_2EMCLAURIN__EXP__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
? [V2t: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V2t ) @ ( c_2Ereal_2Eabs @ V0x ) )
& ( ( c_2Etransc_2Eexp @ V0x )
= ( c_2Erealax_2Ereal__add
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1n )
@ ^ [V3m: tyop_2Enum_2Enum] : ( c_2Ereal_2E_2F @ ( c_2Ereal_2Epow @ V0x @ V3m ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V3m ) ) ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( c_2Etransc_2Eexp @ V2t ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2EFACT @ V1n ) ) ) @ ( c_2Ereal_2Epow @ V0x @ V1n ) ) ) ) ) ).
thf(thm_2Etransc_2EDIFF__LN__COMPOSITE,axiom,
! [V0g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1m: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Elim_2Ediffl @ V0g @ V1m @ V2x )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0g @ V2x ) ) )
=> ( c_2Elim_2Ediffl
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eln @ ( V0g @ V3x ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ ( V0g @ V2x ) ) @ V1m )
@ V2x ) ) ).
thf(thm_2Etransc_2EGEN__RPOW,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
=> ( ( c_2Ereal_2Epow @ V0a @ V1n )
= ( c_2Etransc_2Erpow @ V0a @ ( c_2Ereal_2Ereal__of__num @ V1n ) ) ) ) ).
thf(thm_2Etransc_2ERPOW__SUC__N,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
=> ( ( c_2Etransc_2Erpow @ V0a @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Ereal_2Epow @ V0a @ ( c_2Enum_2ESUC @ V1n ) ) ) ) ).
thf(thm_2Etransc_2ERPOW__0,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
=> ( ( c_2Etransc_2Erpow @ V0a @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Etransc_2ERPOW__1,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
=> ( ( c_2Etransc_2Erpow @ V0a @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= V0a ) ) ).
thf(thm_2Etransc_2EONE__RPOW,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
=> ( ( c_2Etransc_2Erpow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Etransc_2ERPOW__POS__LT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Erpow @ V0a @ V1b ) ) ) ).
thf(thm_2Etransc_2ERPOW__NZ,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
= V0a ) )
=> ( (~)
@ ( ( c_2Etransc_2Erpow @ V0a @ V1b )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Etransc_2ELN__RPOW,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
=> ( ( c_2Etransc_2Eln @ ( c_2Etransc_2Erpow @ V0a @ V1b ) )
= ( c_2Erealax_2Ereal__mul @ V1b @ ( c_2Etransc_2Eln @ V0a ) ) ) ) ).
thf(thm_2Etransc_2ERPOW__ADD,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Erpow @ V0a @ ( c_2Erealax_2Ereal__add @ V1b @ V2c ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Erpow @ V0a @ V1b ) @ ( c_2Etransc_2Erpow @ V0a @ V2c ) ) ) ).
thf(thm_2Etransc_2ERPOW__ADD__MUL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Erpow @ V0a @ ( c_2Erealax_2Ereal__add @ V1b @ V2c ) ) @ ( c_2Etransc_2Erpow @ V0a @ ( c_2Erealax_2Ereal__neg @ V1b ) ) )
= ( c_2Etransc_2Erpow @ V0a @ V2c ) ) ).
thf(thm_2Etransc_2ERPOW__SUB,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Erpow @ V0a @ ( c_2Ereal_2Ereal__sub @ V1b @ V2c ) )
= ( c_2Ereal_2E_2F @ ( c_2Etransc_2Erpow @ V0a @ V1b ) @ ( c_2Etransc_2Erpow @ V0a @ V2c ) ) ) ).
thf(thm_2Etransc_2ERPOW__DIV,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1b ) )
=> ( ( c_2Etransc_2Erpow @ ( c_2Ereal_2E_2F @ V0a @ V1b ) @ V2c )
= ( c_2Ereal_2E_2F @ ( c_2Etransc_2Erpow @ V0a @ V2c ) @ ( c_2Etransc_2Erpow @ V1b @ V2c ) ) ) ) ).
thf(thm_2Etransc_2ERPOW__INV,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
=> ( ( c_2Etransc_2Erpow @ ( c_2Erealax_2Einv @ V0a ) @ V1b )
= ( c_2Erealax_2Einv @ ( c_2Etransc_2Erpow @ V0a @ V1b ) ) ) ) ).
thf(thm_2Etransc_2ERPOW__MUL,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1b ) )
=> ( ( c_2Etransc_2Erpow @ ( c_2Erealax_2Ereal__mul @ V0a @ V1b ) @ V2c )
= ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Erpow @ V0a @ V2c ) @ ( c_2Etransc_2Erpow @ V1b @ V2c ) ) ) ) ).
thf(thm_2Etransc_2ERPOW__RPOW,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
=> ( ( c_2Etransc_2Erpow @ ( c_2Etransc_2Erpow @ V0a @ V1b ) @ V2c )
= ( c_2Etransc_2Erpow @ V0a @ ( c_2Erealax_2Ereal__mul @ V1b @ V2c ) ) ) ) ).
thf(thm_2Etransc_2ERPOW__LT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Etransc_2Erpow @ V0a @ V1b ) @ ( c_2Etransc_2Erpow @ V0a @ V2c ) )
= ( c_2Erealax_2Ereal__lt @ V1b @ V2c ) ) ) ).
thf(thm_2Etransc_2ERPOW__LE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Erpow @ V0a @ V1b ) @ ( c_2Etransc_2Erpow @ V0a @ V2c ) )
= ( c_2Ereal_2Ereal__lte @ V1b @ V2c ) ) ) ).
thf(thm_2Etransc_2EBASE__RPOW__LE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2c )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1b ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Erpow @ V0a @ V1b ) @ ( c_2Etransc_2Erpow @ V2c @ V1b ) )
= ( c_2Ereal_2Ereal__lte @ V0a @ V2c ) ) ) ).
thf(thm_2Etransc_2EBASE__RPOW__LT,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2c )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1b ) )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Etransc_2Erpow @ V0a @ V1b ) @ ( c_2Etransc_2Erpow @ V2c @ V1b ) )
= ( c_2Erealax_2Ereal__lt @ V0a @ V2c ) ) ) ).
thf(thm_2Etransc_2ERPOW__UNIQ__BASE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0a )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2c )
& ( (~)
@ ( ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
= V1b ) )
& ( ( c_2Etransc_2Erpow @ V0a @ V1b )
= ( c_2Etransc_2Erpow @ V2c @ V1b ) ) )
=> ( V0a = V2c ) ) ).
thf(thm_2Etransc_2ERPOW__UNIQ__EXP,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0a )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2c )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1b )
& ( ( c_2Etransc_2Erpow @ V0a @ V1b )
= ( c_2Etransc_2Erpow @ V0a @ V2c ) ) )
=> ( V1b = V2c ) ) ).
thf(thm_2Etransc_2ERPOW__DIV__BASE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1t: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( ( c_2Ereal_2E_2F @ ( c_2Etransc_2Erpow @ V0x @ V1t ) @ V0x )
= ( c_2Etransc_2Erpow @ V0x @ ( c_2Ereal_2Ereal__sub @ V1t @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Etransc_2EDIFF__COMPOSITE__EXP,axiom,
! [V0g: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1m: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Elim_2Ediffl @ V0g @ V1m @ V2x )
=> ( c_2Elim_2Ediffl
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eexp @ ( V0g @ V3x ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Etransc_2Eexp @ ( V0g @ V2x ) ) @ V1m )
@ V2x ) ) ).
thf(thm_2Etransc_2EDIFF__RPOW,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0x )
=> ( c_2Elim_2Ediffl
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Erpow @ V2x @ V1y )
@ ( c_2Erealax_2Ereal__mul @ V1y @ ( c_2Etransc_2Erpow @ V0x @ ( c_2Ereal_2Ereal__sub @ V1y @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
@ V0x ) ) ).
%------------------------------------------------------------------------------