ITP001 Axioms: ITP132^5.ax
%------------------------------------------------------------------------------
% File : ITP132^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : transc^2.ax [Gau20]
% : HL4132^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 295 ( 77 unt; 31 typ; 0 def)
% Number of atoms : 8624 ( 193 equ; 0 cnn)
% Maximal formula atoms : 347 ( 29 avg)
% Number of connectives : 9964 ( 22 ~; 2 |; 164 &;9550 @)
% ( 21 <=>; 205 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 9 avg;9550 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 13 ( 13 >; 0 *; 0 +; 0 <<)
% Number of symbols : 86 ( 85 usr; 74 con; 0-2 aty)
% Number of variables : 498 ( 65 ^ 407 !; 26 ?; 498 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Etransc_2EDint,type,
c_2Etransc_2EDint: $i ).
thf(mem_c_2Etransc_2EDint,axiom,
mem @ c_2Etransc_2EDint @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ) ).
thf(tp_c_2Etransc_2Eacs,type,
c_2Etransc_2Eacs: $i ).
thf(mem_c_2Etransc_2Eacs,axiom,
mem @ c_2Etransc_2Eacs @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Etransc_2Eacs,type,
fo__c_2Etransc_2Eacs: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Etransc_2Eacs,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Etransc_2Eacs @ X0 ) )
= ( ap @ c_2Etransc_2Eacs @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Etransc_2Easn,type,
c_2Etransc_2Easn: $i ).
thf(mem_c_2Etransc_2Easn,axiom,
mem @ c_2Etransc_2Easn @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Etransc_2Easn,type,
fo__c_2Etransc_2Easn: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Etransc_2Easn,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Etransc_2Easn @ X0 ) )
= ( ap @ c_2Etransc_2Easn @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Etransc_2Eatn,type,
c_2Etransc_2Eatn: $i ).
thf(mem_c_2Etransc_2Eatn,axiom,
mem @ c_2Etransc_2Eatn @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Etransc_2Eatn,type,
fo__c_2Etransc_2Eatn: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Etransc_2Eatn,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Etransc_2Eatn @ X0 ) )
= ( ap @ c_2Etransc_2Eatn @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Etransc_2Ecos,type,
c_2Etransc_2Ecos: $i ).
thf(mem_c_2Etransc_2Ecos,axiom,
mem @ c_2Etransc_2Ecos @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Etransc_2Ecos,type,
fo__c_2Etransc_2Ecos: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Etransc_2Ecos,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Etransc_2Ecos @ X0 ) )
= ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Etransc_2Edivision,type,
c_2Etransc_2Edivision: $i ).
thf(mem_c_2Etransc_2Edivision,axiom,
mem @ c_2Etransc_2Edivision @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ bool ) ) ).
thf(tp_c_2Etransc_2Edsize,type,
c_2Etransc_2Edsize: $i ).
thf(mem_c_2Etransc_2Edsize,axiom,
mem @ c_2Etransc_2Edsize @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ty_2Enum_2Enum ) ).
thf(tp_c_2Etransc_2Eexp,type,
c_2Etransc_2Eexp: $i ).
thf(mem_c_2Etransc_2Eexp,axiom,
mem @ c_2Etransc_2Eexp @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Etransc_2Eexp,type,
fo__c_2Etransc_2Eexp: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Etransc_2Eexp,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Etransc_2Eexp @ X0 ) )
= ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Etransc_2Efine,type,
c_2Etransc_2Efine: $i ).
thf(mem_c_2Etransc_2Efine,axiom,
mem @ c_2Etransc_2Efine @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ bool ) ) ).
thf(tp_c_2Etransc_2Egauge,type,
c_2Etransc_2Egauge: $i ).
thf(mem_c_2Etransc_2Egauge,axiom,
mem @ c_2Etransc_2Egauge @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ bool ) ) ).
thf(tp_c_2Etransc_2Eln,type,
c_2Etransc_2Eln: $i ).
thf(mem_c_2Etransc_2Eln,axiom,
mem @ c_2Etransc_2Eln @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Etransc_2Eln,type,
fo__c_2Etransc_2Eln: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Etransc_2Eln,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Etransc_2Eln @ X0 ) )
= ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Etransc_2Epi,type,
c_2Etransc_2Epi: $i ).
thf(mem_c_2Etransc_2Epi,axiom,
mem @ c_2Etransc_2Epi @ ty_2Erealax_2Ereal ).
thf(stp_fo_c_2Etransc_2Epi,type,
fo__c_2Etransc_2Epi: tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Etransc_2Epi,axiom,
( ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi )
= c_2Etransc_2Epi ) ).
thf(tp_c_2Etransc_2Eroot,type,
c_2Etransc_2Eroot: $i ).
thf(mem_c_2Etransc_2Eroot,axiom,
mem @ c_2Etransc_2Eroot @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ).
thf(stp_fo_c_2Etransc_2Eroot,type,
fo__c_2Etransc_2Eroot: tp__ty_2Enum_2Enum > tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Etransc_2Eroot,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Etransc_2Eroot @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Etransc_2Eroot @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Erealax_2Ereal @ X1 ) ) ) ).
thf(tp_c_2Etransc_2Erpow,type,
c_2Etransc_2Erpow: $i ).
thf(mem_c_2Etransc_2Erpow,axiom,
mem @ c_2Etransc_2Erpow @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ).
thf(stp_fo_c_2Etransc_2Erpow,type,
fo__c_2Etransc_2Erpow: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Etransc_2Erpow,axiom,
! [X0: tp__ty_2Erealax_2Ereal,X1: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Etransc_2Erpow @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) @ ( inj__ty_2Erealax_2Ereal @ X1 ) ) ) ).
thf(tp_c_2Etransc_2Ersum,type,
c_2Etransc_2Ersum: $i ).
thf(mem_c_2Etransc_2Ersum,axiom,
mem @ c_2Etransc_2Ersum @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ) ).
thf(tp_c_2Etransc_2Esin,type,
c_2Etransc_2Esin: $i ).
thf(mem_c_2Etransc_2Esin,axiom,
mem @ c_2Etransc_2Esin @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Etransc_2Esin,type,
fo__c_2Etransc_2Esin: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Etransc_2Esin,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Etransc_2Esin @ X0 ) )
= ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Etransc_2Esqrt,type,
c_2Etransc_2Esqrt: $i ).
thf(mem_c_2Etransc_2Esqrt,axiom,
mem @ c_2Etransc_2Esqrt @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Etransc_2Esqrt,type,
fo__c_2Etransc_2Esqrt: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Etransc_2Esqrt,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Etransc_2Esqrt @ X0 ) )
= ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Etransc_2Etan,type,
c_2Etransc_2Etan: $i ).
thf(mem_c_2Etransc_2Etan,axiom,
mem @ c_2Etransc_2Etan @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Etransc_2Etan,type,
fo__c_2Etransc_2Etan: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Etransc_2Etan,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Etransc_2Etan @ X0 ) )
= ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Etransc_2Etdiv,type,
c_2Etransc_2Etdiv: $i ).
thf(mem_c_2Etransc_2Etdiv,axiom,
mem @ c_2Etransc_2Etdiv @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ bool ) ) ).
thf(ax_thm_2Etransc_2Eexp,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ c_2Eseq_2Esuminf
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] :
( ap
@ ( ap @ c_2Erealax_2Ereal__mul
@ ( ap
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V2n ) ) ) )
@ V1n ) )
@ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1n ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Ecos,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ c_2Eseq_2Esuminf
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] :
( ap
@ ( ap @ c_2Erealax_2Ereal__mul
@ ( ap
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Earithmetic_2EEVEN @ V2n ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ V2n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V2n ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
@ V1n ) )
@ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1n ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Esin,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ c_2Eseq_2Esuminf
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] :
( ap
@ ( ap @ c_2Erealax_2Ereal__mul
@ ( ap
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Earithmetic_2EEVEN @ V2n ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V2n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V2n ) ) ) ) )
@ V1n ) )
@ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1n ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__CONVERGES,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] :
( ap
@ ( ap @ c_2Erealax_2Ereal__mul
@ ( ap
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V2n ) ) ) )
@ V1n ) )
@ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1n ) ) ) )
@ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__CONVERGES,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] :
( ap
@ ( ap @ c_2Erealax_2Ereal__mul
@ ( ap
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Earithmetic_2EEVEN @ V2n ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V2n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V2n ) ) ) ) )
@ V1n ) )
@ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1n ) ) ) )
@ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__CONVERGES,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] :
( ap
@ ( ap @ c_2Erealax_2Ereal__mul
@ ( ap
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Earithmetic_2EEVEN @ V2n ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ V2n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V2n ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
@ V1n ) )
@ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1n ) ) ) )
@ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__FDIFF,axiom,
( ( ap @ c_2Epowser_2Ediffs
@ ( lam @ ty_2Enum_2Enum
@ ^ [V0n: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V0n ) ) ) ) )
= ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V1n ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__FDIFF,axiom,
( ( ap @ c_2Epowser_2Ediffs
@ ( lam @ ty_2Enum_2Enum
@ ^ [V0n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Earithmetic_2EEVEN @ V0n ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V0n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V0n ) ) ) ) ) )
= ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Earithmetic_2EEVEN @ V1n ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ V1n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V1n ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__FDIFF,axiom,
( ( ap @ c_2Epowser_2Ediffs
@ ( lam @ ty_2Enum_2Enum
@ ^ [V0n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Earithmetic_2EEVEN @ V0n ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ V0n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V0n ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
= ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] :
( ap @ c_2Erealax_2Ereal__neg
@ ( ap
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Earithmetic_2EEVEN @ V2n ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V2n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V2n ) ) ) ) )
@ V1n ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__NEGLEMMA,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ c_2Eseq_2Esuminf
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] :
( ap @ c_2Erealax_2Ereal__neg
@ ( ap
@ ( ap @ c_2Erealax_2Ereal__mul
@ ( ap
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Earithmetic_2EEVEN @ V2n ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V2n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V2n ) ) ) ) )
@ V1n ) )
@ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1n ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__EXP,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ c_2Etransc_2Eexp ) @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__SIN,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ c_2Etransc_2Esin ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__COS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ c_2Etransc_2Ecos ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__COMPOSITE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal,V3g: $i] :
( ( mem @ V3g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V4m: tp__ty_2Erealax_2Ereal,V5n: tp__ty_2Enum_2Enum] :
( ( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ V0f @ V6x ) ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V3g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ V0f @ V7x ) ) @ ( ap @ V3g @ V7x ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V3g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V8x ) ) @ ( ap @ V3g @ V8x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V3g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V9x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V9x ) ) @ ( ap @ V3g @ V9x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V3g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V10x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V10x ) ) @ ( ap @ V3g @ V10x ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V11x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V11x ) ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V3g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V12x: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ V3g @ V12x ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V3g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V13x: $i] : ( ap @ c_2Etransc_2Eexp @ ( ap @ V3g @ V13x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Eexp @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V3g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V14x: $i] : ( ap @ c_2Etransc_2Esin @ ( ap @ V3g @ V14x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Ecos @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V3g ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V15x: $i] : ( ap @ c_2Etransc_2Ecos @ ( ap @ V3g @ V15x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Etransc_2Esin @ ( ap @ V3g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__0,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__LE__X,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__LT__1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__ADD__MUL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Eexp @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ c_2Etransc_2Eexp @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__NEG__MUL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Eexp @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__NEG__MUL2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Eexp @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__ADD,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__POS__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__NZ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__POS__LT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__N,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__SUB,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__MONO__IMP,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__MONO__LT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__MONO__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__INJ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( V0x = V1y ) ) ).
thf(conj_thm_2Etransc_2EEXP__TOTAL__LEMMA,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
=> ? [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= V0y ) ) ) ).
thf(conj_thm_2Etransc_2EEXP__TOTAL,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
=> ? [V1x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= V0y ) ) ).
thf(ax_thm_2Etransc_2Eln,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1u: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Eexp @ V1u ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ELN__EXP,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eln @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= V0x ) ).
thf(conj_thm_2Etransc_2EEXP__LN,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= V0x )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ELN__MUL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eln @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ELN__INJ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( V0x = V1y ) ) ) ).
thf(conj_thm_2Etransc_2ELN__1,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eln @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2ELN__INV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eln @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ELN__DIV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eln @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ELN__MONO__LT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ELN__MONO__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ELN__POW,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eln @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ELN__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Etransc_2Eln @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ELN__LT__X,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ELN__POS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Eroot,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2u: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2u ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ V2u ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Esqrt,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2EROOT__LT__LEMMA,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Eexp @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= V1x ) ) ).
thf(conj_thm_2Etransc_2EROOT__LN,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EROOT__0,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2EROOT__1,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EROOT__POS__LT,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EROOT__POW__POS,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= V1x ) ) ).
thf(conj_thm_2Etransc_2EPOW__ROOT__POS,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= V1x ) ) ).
thf(conj_thm_2Etransc_2EROOT__POS,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EROOT__POS__UNIQ,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= V1x ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= V2y ) ) ).
thf(conj_thm_2Etransc_2EROOT__MUL,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EROOT__INV,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Einv @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EROOT__DIV,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EROOT__MONO__LE,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Eroot @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESQRT__0,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2ESQRT__1,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESQRT__POS__LT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESQRT__POS__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESQRT__POW2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= V0x )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ESQRT__POW__2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= V0x ) ) ).
thf(conj_thm_2Etransc_2EPOW__2__SQRT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= V0x ) ) ).
thf(conj_thm_2Etransc_2ESQRT__POS__UNIQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= V0x ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= V1y ) ) ).
thf(conj_thm_2Etransc_2ESQRT__MUL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESQRT__INV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESQRT__DIV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESQRT__MONO__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESQRT__EVEN__POW2,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EREAL__DIV__SQRT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESQRT__EQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= V1y )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
=> ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__0,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__0,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__CIRCLE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__BOUND,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__BOUNDS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__BOUND,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__BOUNDS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__COS__ADD,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Etransc_2Esin @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Etransc_2Ecos @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__COS__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__ADD,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__ADD,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__DOUBLE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__DOUBLE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__PAIRED,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V1n ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
@ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__POS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__PAIRED,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V1n ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V1n ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V1n ) ) ) ) )
@ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__2,axiom,
p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__ISZERO,axiom,
( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V0x ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Epi,axiom,
( fo__c_2Etransc_2Epi
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V0x ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EPI2,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V0x ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__PI2,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2EPI2__BOUNDS,axiom,
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EPI__POS,axiom,
p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ).
thf(conj_thm_2Etransc_2ESIN__PI2,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__PI,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__PI,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__COS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__SIN,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__PERIODIC__PI,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__PERIODIC__PI,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__PERIODIC,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__PERIODIC,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__NPI,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__NPI,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__POS__PI2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__POS__PI2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__POS__PI,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__POS__PI,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__POS__PI2__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__POS__PI__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__POS__PI2__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__POS__PI__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__TOTAL,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__TOTAL,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ V1x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Esin @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__ZERO__LEMMA,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ? [V1n: tp__ty_2Enum_2Enum] :
( ~ ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__ZERO__LEMMA,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ? [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__ZERO,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ? [V1n: tp__ty_2Enum_2Enum] :
( ~ ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
| ? [V2n: tp__ty_2Enum_2Enum] :
( ~ ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__ZERO,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ? [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
| ? [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Etan,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ETAN__0,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Etan @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2ETAN__PI,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2ETAN__NPI,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Etan @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2ETAN__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Etan @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ETAN__PERIODIC,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Etan @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ETAN__ADD,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Etan @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ETAN__DOUBLE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Etan @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ETAN__POS__PI2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__TAN,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ c_2Etransc_2Etan ) @ ( ap @ c_2Erealax_2Einv @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2ETAN__TOTAL__LEMMA,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
=> ? [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ETAN__TOTAL__POS,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
=> ? [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
= V0y ) ) ) ).
thf(conj_thm_2Etransc_2ETAN__TOTAL,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ V1x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Etan @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Easn,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Easn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ V1x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Esin @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Eacs,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eacs @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Eatn,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eatn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ V1x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V1x ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Etan @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EASN,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ c_2Etransc_2Easn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Etransc_2Easn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Etransc_2Easn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
= V0y ) ) ) ).
thf(conj_thm_2Etransc_2EASN__SIN,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Etransc_2Easn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
= V0y ) ) ).
thf(conj_thm_2Etransc_2EASN__BOUNDS,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ c_2Etransc_2Easn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Etransc_2Easn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EASN__BOUNDS__LT,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ c_2Etransc_2Easn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Etransc_2Easn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__ASN,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Easn @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= V0x ) ) ).
thf(conj_thm_2Etransc_2EACS,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Eacs @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Etransc_2Eacs @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Etransc_2Eacs @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
= V0y ) ) ) ).
thf(conj_thm_2Etransc_2EACS__COS,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Etransc_2Eacs @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
= V0y ) ) ).
thf(conj_thm_2Etransc_2EACS__BOUNDS,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Eacs @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Etransc_2Eacs @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EACS__BOUNDS__LT,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Eacs @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Etransc_2Eacs @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__ACS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eacs @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= V0x ) ) ).
thf(conj_thm_2Etransc_2EATN,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ c_2Etransc_2Eatn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Etransc_2Eatn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Etan @ ( ap @ c_2Etransc_2Eatn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
= V0y ) ) ).
thf(conj_thm_2Etransc_2EATN__TAN,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Etan @ ( ap @ c_2Etransc_2Eatn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
= V0y ) ).
thf(conj_thm_2Etransc_2EATN__BOUNDS,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ c_2Etransc_2Eatn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Etransc_2Eatn @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ETAN__ATN,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eatn @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= V0x ) ) ).
thf(conj_thm_2Etransc_2ETAN__SEC,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Etan @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__COS__SQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__SIN__SQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__ATN__NZ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Etransc_2Eatn @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__ASN__NZ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Etransc_2Easn @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__ACS__NZ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Etransc_2Eacs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ECOS__SIN__SQRT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ESIN__COS__SQRT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__LN,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ c_2Etransc_2Eln ) @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__ASN__LEMMA,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ c_2Etransc_2Easn ) @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Etransc_2Easn @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__ASN,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ c_2Etransc_2Easn ) @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__ACS__LEMMA,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ c_2Etransc_2Eacs ) @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Etransc_2Eacs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__ACS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ c_2Etransc_2Eacs ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__ATN,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ c_2Etransc_2Eatn ) @ ( ap @ c_2Erealax_2Einv @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ).
thf(ax_thm_2Etransc_2Edivision,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2D: $i] :
( ( mem @ V2D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2D ) )
<=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2D @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= V0a )
& ? [V3N: tp__ty_2Enum_2Enum] :
( ! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( inj__ty_2Enum_2Enum @ V3N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V2D @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V2D @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) )
& ! [V5n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ ( inj__ty_2Enum_2Enum @ V3N ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2D @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
= V1b ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Edsize,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Etransc_2Edsize @ V0D ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1N: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V2n ) @ V1N ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0D @ V2n ) ) @ ( ap @ V0D @ ( ap @ c_2Enum_2ESUC @ V2n ) ) ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ V3n ) @ V1N ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0D @ V3n ) ) @ ( ap @ V0D @ V1N ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Etdiv,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2D: $i] :
( ( mem @ V2D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3p: $i] :
( ( mem @ V3p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V2D ) @ V3p ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2D ) )
& ! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V2D @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V3p @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V3p @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V2D @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Egauge,axiom,
! [V0E: $i] :
( ( mem @ V0E @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Etransc_2Egauge @ V0E ) @ V1g ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ V0E @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Efine,axiom,
! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1D: $i] :
( ( mem @ V1D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V0g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V1D ) @ V2p ) ) )
<=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Etransc_2Edsize @ V1D ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1D @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ V1D @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ V0g @ ( ap @ V2p @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Ersum,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V0D ) @ V1p ) ) @ V2f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Etransc_2Edsize @ V0D ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2f @ ( ap @ V1p @ V3n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0D @ ( ap @ c_2Enum_2ESUC @ V3n ) ) ) @ ( ap @ V0D @ V3n ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etransc_2EDint,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3k: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V3k ) ) )
<=> ! [V4e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
=> ? [V5g: $i] :
( ( mem @ V5g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p
@ ( ap
@ ( ap @ c_2Etransc_2Egauge
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V6x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V6x ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
@ V5g ) )
& ! [V7D: $i] :
( ( mem @ V7D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V8p: $i] :
( ( mem @ V8p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V7D ) @ V8p ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V5g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V7D ) @ V8p ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V7D ) @ V8p ) ) @ V2f ) ) @ ( inj__ty_2Erealax_2Ereal @ V3k ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__0,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( V0a = V1b )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ c_2Etransc_2Edsize
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V2n ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
= fo__c_2Enum_2E0 ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__1,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ c_2Etransc_2Edsize
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V2n ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__SINGLE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V2n ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__LHS,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0D ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0D @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= V1a ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__THM,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0D ) )
<=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0D @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= V1a )
& ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Etransc_2Edsize @ V0D ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0D @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V0D @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ c_2Etransc_2Edsize @ V0D ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0D @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
= V2b ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__RHS,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0D ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0D @ ( ap @ c_2Etransc_2Edsize @ V0D ) ) )
= V2b ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__LT__GEN,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0D ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ c_2Etransc_2Edsize @ V0D ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0D @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V0D @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__LT,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0D ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Etransc_2Edsize @ V0D ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0D @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0D @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__LE,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0D ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__GT,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0D ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Etransc_2Edsize @ V0D ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0D @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V0D @ ( ap @ c_2Etransc_2Edsize @ V0D ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__EQ,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0D ) )
=> ( ( V1a = V2b )
<=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Etransc_2Edsize @ V0D ) )
= fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__LBOUND,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0D ) )
=> ! [V3r: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ V0D @ ( inj__ty_2Enum_2Enum @ V3r ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__LBOUND__LT,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0D ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Etransc_2Edsize @ V0D ) )
!= fo__c_2Enum_2E0 ) )
=> ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ V0D @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__UBOUND,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0D ) )
=> ! [V3r: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0D @ ( inj__ty_2Enum_2Enum @ V3r ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__UBOUND__LT,axiom,
! [V0D: $i] :
( ( mem @ V0D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0D ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Etransc_2Edsize @ V0D ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0D @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__APPEND,axiom,
! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ? [V4D1: $i] :
( ( mem @ V4D1 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ? [V5p1: $i] :
( ( mem @ V5p1 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V4D1 ) @ V5p1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V0g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V4D1 ) @ V5p1 ) ) ) ) )
& ? [V6D2: $i] :
( ( mem @ V6D2 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ? [V7p2: $i] :
( ( mem @ V7p2 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6D2 ) @ V7p2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V0g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6D2 ) @ V7p2 ) ) ) ) ) )
=> ? [V8D: $i] :
( ( mem @ V8D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ? [V9p: $i] :
( ( mem @ V9p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8D ) @ V9p ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V0g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8D ) @ V9p ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIVISION__EXISTS,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
& ( p
@ ( ap
@ ( ap @ c_2Etransc_2Egauge
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V3x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V3x ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
@ V2g ) ) )
=> ? [V4D: $i] :
( ( mem @ V4D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ? [V5p: $i] :
( ( mem @ V5p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V4D ) @ V5p ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V2g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V4D ) @ V5p ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EGAUGE__MIN,axiom,
! [V0E: $i] :
( ( mem @ V0E @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1g1: $i] :
( ( mem @ V1g1 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2g2: $i] :
( ( mem @ V2g2 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Egauge @ V0E ) @ V1g1 ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Egauge @ V0E ) @ V2g2 ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Etransc_2Egauge @ V0E )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V1g1 @ V3x ) ) @ ( ap @ V2g2 @ V3x ) ) ) @ ( ap @ V1g1 @ V3x ) ) @ ( ap @ V2g2 @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EFINE__MIN,axiom,
! [V0g1: $i] :
( ( mem @ V0g1 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g2: $i] :
( ( mem @ V1g2 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2D: $i] :
( ( mem @ V2D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3p: $i] :
( ( mem @ V3p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p
@ ( ap
@ ( ap @ c_2Etransc_2Efine
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0g1 @ V4x ) ) @ ( ap @ V1g2 @ V4x ) ) ) @ ( ap @ V0g1 @ V4x ) ) @ ( ap @ V1g2 @ V4x ) ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V2D ) @ V3p ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V0g1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V2D ) @ V3p ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V1g2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V2D ) @ V3p ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDINT__UNIQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3k1: tp__ty_2Erealax_2Ereal,V4k2: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V3k1 ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V4k2 ) ) ) )
=> ( V3k1 = V4k2 ) ) ) ).
thf(conj_thm_2Etransc_2EINTEGRAL__NULL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EFTC1,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1f_27: $i] :
( ( mem @ V1f_27 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( ap @ V1f_27 @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1f_27 ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EMCLAURIN,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1diff: $i] :
( ( mem @ V1diff @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V2h: tp__ty_2Erealax_2Ereal,V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
& ( ( ap @ V1diff @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= V0f )
& ! [V4m: tp__ty_2Enum_2Enum,V5t: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5t ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5t ) ) @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ ( ap @ V1diff @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) @ ( ap @ ( ap @ V1diff @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5t ) ) ) ) )
=> ? [V6t: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6t ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V6t ) ) @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Erealax_2Ereal__add
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7m: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ V1diff @ V7m ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V7m ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) @ V7m ) ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ V1diff @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Erealax_2Ereal @ V6t ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EMCLAURIN__NEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1diff: $i] :
( ( mem @ V1diff @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V2h: tp__ty_2Erealax_2Ereal,V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
& ( ( ap @ V1diff @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= V0f )
& ! [V4m: tp__ty_2Enum_2Enum,V5t: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) @ ( inj__ty_2Erealax_2Ereal @ V5t ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5t ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ ( ap @ V1diff @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) @ ( ap @ ( ap @ V1diff @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5t ) ) ) ) )
=> ? [V6t: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) @ ( inj__ty_2Erealax_2Ereal @ V6t ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V6t ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Erealax_2Ereal__add
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7m: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ V1diff @ V7m ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V7m ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) @ V7m ) ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ V1diff @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Erealax_2Ereal @ V6t ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EMCLAURIN__ALL__LT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1diff: $i] :
( ( mem @ V1diff @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ( ( ( ( ap @ V1diff @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= V0f )
& ! [V2m: tp__ty_2Enum_2Enum,V3x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ ( ap @ V1diff @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ ( ap @ V1diff @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
=> ! [V4x: tp__ty_2Erealax_2Ereal,V5n: tp__ty_2Enum_2Enum] :
( ( ( V4x
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) )
=> ? [V6t: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V6t ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V6t ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Erealax_2Ereal__add
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7m: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ V1diff @ V7m ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V7m ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V7m ) ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ V1diff @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ ( inj__ty_2Erealax_2Ereal @ V6t ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EMCLAURIN__ZERO,axiom,
! [V0diff: $i] :
( ( mem @ V0diff @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( V2x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3m: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ V0diff @ V3m ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V3m ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V3m ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ V0diff @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EMCLAURIN__ALL__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1diff: $i] :
( ( mem @ V1diff @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ( ( ( ( ap @ V1diff @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= V0f )
& ! [V2m: tp__ty_2Enum_2Enum,V3x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ ( ap @ V1diff @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ ( ap @ V1diff @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
=> ! [V4x: tp__ty_2Erealax_2Ereal,V5n: tp__ty_2Enum_2Enum] :
? [V6t: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V6t ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Erealax_2Ereal__add
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7m: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ V1diff @ V7m ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V7m ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V7m ) ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ V1diff @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ ( inj__ty_2Erealax_2Ereal @ V6t ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EMCLAURIN__EXP__LT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( ( V0x
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ? [V2t: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2t ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2t ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Erealax_2Ereal__add
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3m: $i] : ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V3m ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V3m ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V2t ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EMCLAURIN__EXP__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
? [V2t: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2t ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Erealax_2Ereal__add
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3m: $i] : ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V3m ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ V3m ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Etransc_2Eexp @ ( inj__ty_2Erealax_2Ereal @ V2t ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2EFACT @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__LN__COMPOSITE,axiom,
! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0g ) @ ( inj__ty_2Erealax_2Ereal @ V1m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ c_2Etransc_2Eln @ ( ap @ V0g @ V3x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( ap @ V0g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ).
thf(ax_thm_2Etransc_2Erpow__def,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eexp @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EGEN__RPOW,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__SUC__N,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__0,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__1,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= V0a ) ) ).
thf(conj_thm_2Etransc_2EONE__RPOW,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__POS__LT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__NZ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
!= V0a )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ELN__RPOW,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Eln @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ c_2Etransc_2Eln @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__ADD,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__ADD__MUL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__SUB,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__DIV,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__INV,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Einv @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__MUL,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__RPOW,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__LT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__LE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EBASE__RPOW__LE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EBASE__RPOW__LT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ).
thf(conj_thm_2Etransc_2ERPOW__UNIQ__BASE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
!= V1b )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
=> ( V0a = V2c ) ) ).
thf(conj_thm_2Etransc_2ERPOW__UNIQ__EXP,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) )
=> ( V1b = V2c ) ) ).
thf(conj_thm_2Etransc_2ERPOW__DIV__BASE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1t: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1t ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1t ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__COMPOSITE__EXP,axiom,
! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0g ) @ ( inj__ty_2Erealax_2Ereal @ V1m ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ c_2Etransc_2Eexp @ ( ap @ V0g @ V3x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Etransc_2Eexp @ ( ap @ V0g @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1m ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ).
thf(conj_thm_2Etransc_2EDIFF__RPOW,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Etransc_2Erpow @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ ( ap @ c_2Etransc_2Erpow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
%------------------------------------------------------------------------------