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 : 389 ( 139 unt; 78 typ; 0 def)
% Number of atoms : 8386 ( 240 equ)
% Maximal formula atoms : 28 ( 21 avg)
% Number of connectives : 443 ( 22 ~; 2 |; 164 &)
% ( 21 <=>; 234 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 4 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 7654 (7654 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 68 ( 48 >; 20 *; 0 +; 0 <<)
% Number of predicates : 54 ( 53 usr; 38 prp; 0-2 aty)
% Number of functors : 78 ( 78 usr; 30 con; 0-3 aty)
% Number of variables : 535 ( 509 !; 26 ?; 535 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Etransc_2EDint,type,
c_2Etransc_2EDint: $i ).
tff(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)))) ).
tff(tp_c_2Etransc_2Eacs,type,
c_2Etransc_2Eacs: $i ).
tff(mem_c_2Etransc_2Eacs,axiom,
mem(c_2Etransc_2Eacs,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
tff(stp_fo_c_2Etransc_2Eacs,type,
fo__c_2Etransc_2Eacs: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
tff(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)) ) ).
tff(tp_c_2Etransc_2Easn,type,
c_2Etransc_2Easn: $i ).
tff(mem_c_2Etransc_2Easn,axiom,
mem(c_2Etransc_2Easn,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
tff(stp_fo_c_2Etransc_2Easn,type,
fo__c_2Etransc_2Easn: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
tff(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)) ) ).
tff(tp_c_2Etransc_2Eatn,type,
c_2Etransc_2Eatn: $i ).
tff(mem_c_2Etransc_2Eatn,axiom,
mem(c_2Etransc_2Eatn,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
tff(stp_fo_c_2Etransc_2Eatn,type,
fo__c_2Etransc_2Eatn: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
tff(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)) ) ).
tff(tp_c_2Etransc_2Ecos,type,
c_2Etransc_2Ecos: $i ).
tff(mem_c_2Etransc_2Ecos,axiom,
mem(c_2Etransc_2Ecos,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
tff(stp_fo_c_2Etransc_2Ecos,type,
fo__c_2Etransc_2Ecos: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
tff(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)) ) ).
tff(tp_c_2Etransc_2Edivision,type,
c_2Etransc_2Edivision: $i ).
tff(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))) ).
tff(tp_c_2Etransc_2Edsize,type,
c_2Etransc_2Edsize: $i ).
tff(mem_c_2Etransc_2Edsize,axiom,
mem(c_2Etransc_2Edsize,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),ty_2Enum_2Enum)) ).
tff(tp_c_2Etransc_2Eexp,type,
c_2Etransc_2Eexp: $i ).
tff(mem_c_2Etransc_2Eexp,axiom,
mem(c_2Etransc_2Eexp,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
tff(stp_fo_c_2Etransc_2Eexp,type,
fo__c_2Etransc_2Eexp: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
tff(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)) ) ).
tff(tp_c_2Etransc_2Efine,type,
c_2Etransc_2Efine: $i ).
tff(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))) ).
tff(tp_c_2Etransc_2Egauge,type,
c_2Etransc_2Egauge: $i ).
tff(mem_c_2Etransc_2Egauge,axiom,
mem(c_2Etransc_2Egauge,arr(arr(ty_2Erealax_2Ereal,bool),arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),bool))) ).
tff(tp_c_2Etransc_2Eln,type,
c_2Etransc_2Eln: $i ).
tff(mem_c_2Etransc_2Eln,axiom,
mem(c_2Etransc_2Eln,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
tff(stp_fo_c_2Etransc_2Eln,type,
fo__c_2Etransc_2Eln: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
tff(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)) ) ).
tff(tp_c_2Etransc_2Epi,type,
c_2Etransc_2Epi: $i ).
tff(mem_c_2Etransc_2Epi,axiom,
mem(c_2Etransc_2Epi,ty_2Erealax_2Ereal) ).
tff(stp_fo_c_2Etransc_2Epi,type,
fo__c_2Etransc_2Epi: tp__ty_2Erealax_2Ereal ).
tff(stp_eq_fo_c_2Etransc_2Epi,axiom,
inj__ty_2Erealax_2Ereal(fo__c_2Etransc_2Epi) = c_2Etransc_2Epi ).
tff(tp_c_2Etransc_2Eroot,type,
c_2Etransc_2Eroot: $i ).
tff(mem_c_2Etransc_2Eroot,axiom,
mem(c_2Etransc_2Eroot,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
tff(stp_fo_c_2Etransc_2Eroot,type,
fo__c_2Etransc_2Eroot: ( tp__ty_2Enum_2Enum * tp__ty_2Erealax_2Ereal ) > tp__ty_2Erealax_2Ereal ).
tff(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)) ) ).
tff(tp_c_2Etransc_2Erpow,type,
c_2Etransc_2Erpow: $i ).
tff(mem_c_2Etransc_2Erpow,axiom,
mem(c_2Etransc_2Erpow,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
tff(stp_fo_c_2Etransc_2Erpow,type,
fo__c_2Etransc_2Erpow: ( tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > tp__ty_2Erealax_2Ereal ).
tff(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)) ) ).
tff(tp_c_2Etransc_2Ersum,type,
c_2Etransc_2Ersum: $i ).
tff(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))) ).
tff(tp_c_2Etransc_2Esin,type,
c_2Etransc_2Esin: $i ).
tff(mem_c_2Etransc_2Esin,axiom,
mem(c_2Etransc_2Esin,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
tff(stp_fo_c_2Etransc_2Esin,type,
fo__c_2Etransc_2Esin: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
tff(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)) ) ).
tff(tp_c_2Etransc_2Esqrt,type,
c_2Etransc_2Esqrt: $i ).
tff(mem_c_2Etransc_2Esqrt,axiom,
mem(c_2Etransc_2Esqrt,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
tff(stp_fo_c_2Etransc_2Esqrt,type,
fo__c_2Etransc_2Esqrt: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
tff(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)) ) ).
tff(tp_c_2Etransc_2Etan,type,
c_2Etransc_2Etan: $i ).
tff(mem_c_2Etransc_2Etan,axiom,
mem(c_2Etransc_2Etan,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
tff(stp_fo_c_2Etransc_2Etan,type,
fo__c_2Etransc_2Etan: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
tff(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)) ) ).
tff(tp_c_2Etransc_2Etdiv,type,
c_2Etransc_2Etdiv: $i ).
tff(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))) ).
tff(lamtp_f2870,type,
f2870: $i ).
tff(lameq_f2870,axiom,
! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2870,inj__ty_2Enum_2Enum(V2n)) = ap(c_2Erealax_2Einv,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2EFACT,inj__ty_2Enum_2Enum(V2n)))) ) ).
tff(lamtp_f2871,type,
f2871: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2871,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] : ( ap(f2871(V0x),inj__ty_2Enum_2Enum(V1n)) = ap(ap(c_2Erealax_2Ereal__mul,ap(f2870,inj__ty_2Enum_2Enum(V1n))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V1n))) ) ).
tff(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,f2871(V0x))) ) ).
tff(lamtp_f2872,type,
f2872: $i ).
tff(lameq_f2872,axiom,
! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2872,inj__ty_2Enum_2Enum(V2n)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(c_2Earithmetic_2EEVEN,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(V2n))))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ).
tff(lamtp_f2873,type,
f2873: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2873,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] : ( ap(f2873(V0x),inj__ty_2Enum_2Enum(V1n)) = ap(ap(c_2Erealax_2Ereal__mul,ap(f2872,inj__ty_2Enum_2Enum(V1n))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V1n))) ) ).
tff(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,f2873(V0x))) ) ).
tff(lamtp_f2874,type,
f2874: $i ).
tff(lameq_f2874,axiom,
! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2874,inj__ty_2Enum_2Enum(V2n)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(c_2Earithmetic_2EEVEN,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(V2n))))) ) ).
tff(lamtp_f2875,type,
f2875: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2875,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] : ( ap(f2875(V0x),inj__ty_2Enum_2Enum(V1n)) = ap(ap(c_2Erealax_2Ereal__mul,ap(f2874,inj__ty_2Enum_2Enum(V1n))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V1n))) ) ).
tff(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,f2875(V0x))) ) ).
tff(conj_thm_2Etransc_2EEXP__CONVERGES,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Eseq_2Esums,f2871(V0x)),ap(c_2Etransc_2Eexp,inj__ty_2Erealax_2Ereal(V0x)))) ).
tff(conj_thm_2Etransc_2ESIN__CONVERGES,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Eseq_2Esums,f2875(V0x)),ap(c_2Etransc_2Esin,inj__ty_2Erealax_2Ereal(V0x)))) ).
tff(conj_thm_2Etransc_2ECOS__CONVERGES,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Eseq_2Esums,f2873(V0x)),ap(c_2Etransc_2Ecos,inj__ty_2Erealax_2Ereal(V0x)))) ).
tff(lamtp_f2876,type,
f2876: $i ).
tff(lameq_f2876,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( ap(f2876,inj__ty_2Enum_2Enum(V0n)) = ap(c_2Erealax_2Einv,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2EFACT,inj__ty_2Enum_2Enum(V0n)))) ) ).
tff(lamtp_f2877,type,
f2877: $i ).
tff(lameq_f2877,axiom,
! [V1n: tp__ty_2Enum_2Enum] : ( ap(f2877,inj__ty_2Enum_2Enum(V1n)) = ap(c_2Erealax_2Einv,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2EFACT,inj__ty_2Enum_2Enum(V1n)))) ) ).
tff(conj_thm_2Etransc_2EEXP__FDIFF,axiom,
ap(c_2Epowser_2Ediffs,f2876) = f2877 ).
tff(lamtp_f2878,type,
f2878: $i ).
tff(lameq_f2878,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( ap(f2878,inj__ty_2Enum_2Enum(V0n)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(c_2Earithmetic_2EEVEN,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(V0n))))) ) ).
tff(lamtp_f2879,type,
f2879: $i ).
tff(lameq_f2879,axiom,
! [V1n: tp__ty_2Enum_2Enum] : ( ap(f2879,inj__ty_2Enum_2Enum(V1n)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(c_2Earithmetic_2EEVEN,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(V1n))))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ).
tff(conj_thm_2Etransc_2ESIN__FDIFF,axiom,
ap(c_2Epowser_2Ediffs,f2878) = f2879 ).
tff(lamtp_f2880,type,
f2880: $i ).
tff(lameq_f2880,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( ap(f2880,inj__ty_2Enum_2Enum(V0n)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(c_2Earithmetic_2EEVEN,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(V0n))))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ).
tff(lamtp_f2881,type,
f2881: $i ).
tff(lameq_f2881,axiom,
! [V1n: tp__ty_2Enum_2Enum] : ( ap(f2881,inj__ty_2Enum_2Enum(V1n)) = ap(c_2Erealax_2Ereal__neg,ap(f2874,inj__ty_2Enum_2Enum(V1n))) ) ).
tff(conj_thm_2Etransc_2ECOS__FDIFF,axiom,
ap(c_2Epowser_2Ediffs,f2880) = f2881 ).
tff(lamtp_f2882,type,
f2882: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2882,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] : ( ap(f2882(V0x),inj__ty_2Enum_2Enum(V1n)) = ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Erealax_2Ereal__mul,ap(f2874,inj__ty_2Enum_2Enum(V1n))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V1n)))) ) ).
tff(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,f2882(V0x))) ) ).
tff(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))) ).
tff(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))) ).
tff(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))) ).
tff(lamtp_f2883,type,
f2883: $i > $i ).
tff(lameq_f2883,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V6x: tp__ty_2Erealax_2Ereal] : ( ap(f2883(V0f),inj__ty_2Erealax_2Ereal(V6x)) = ap(c_2Erealax_2Einv,ap(V0f,inj__ty_2Erealax_2Ereal(V6x))) ) ) ).
tff(lamtp_f2884,type,
f2884: ( $i * $i ) > $i ).
tff(lameq_f2884,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3g: $i] :
( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V7x: tp__ty_2Erealax_2Ereal] : ( ap(f2884(V0f,V3g),inj__ty_2Erealax_2Ereal(V7x)) = ap(ap(c_2Ereal_2E_2F,ap(V0f,inj__ty_2Erealax_2Ereal(V7x))),ap(V3g,inj__ty_2Erealax_2Ereal(V7x))) ) ) ) ).
tff(lamtp_f2885,type,
f2885: ( $i * $i ) > $i ).
tff(lameq_f2885,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3g: $i] :
( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V8x: tp__ty_2Erealax_2Ereal] : ( ap(f2885(V0f,V3g),inj__ty_2Erealax_2Ereal(V8x)) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,inj__ty_2Erealax_2Ereal(V8x))),ap(V3g,inj__ty_2Erealax_2Ereal(V8x))) ) ) ) ).
tff(lamtp_f2886,type,
f2886: ( $i * $i ) > $i ).
tff(lameq_f2886,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3g: $i] :
( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V9x: tp__ty_2Erealax_2Ereal] : ( ap(f2886(V0f,V3g),inj__ty_2Erealax_2Ereal(V9x)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Erealax_2Ereal(V9x))),ap(V3g,inj__ty_2Erealax_2Ereal(V9x))) ) ) ) ).
tff(lamtp_f2887,type,
f2887: ( $i * $i ) > $i ).
tff(lameq_f2887,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3g: $i] :
( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V10x: tp__ty_2Erealax_2Ereal] : ( ap(f2887(V0f,V3g),inj__ty_2Erealax_2Ereal(V10x)) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,inj__ty_2Erealax_2Ereal(V10x))),ap(V3g,inj__ty_2Erealax_2Ereal(V10x))) ) ) ) ).
tff(lamtp_f2888,type,
f2888: $i > $i ).
tff(lameq_f2888,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V11x: tp__ty_2Erealax_2Ereal] : ( ap(f2888(V0f),inj__ty_2Erealax_2Ereal(V11x)) = ap(c_2Erealax_2Ereal__neg,ap(V0f,inj__ty_2Erealax_2Ereal(V11x))) ) ) ).
tff(lamtp_f2889,type,
f2889: ( $i * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f2889,axiom,
! [V3g: $i] :
( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V5n: tp__ty_2Enum_2Enum,V12x: tp__ty_2Erealax_2Ereal] : ( ap(f2889(V3g,V5n),inj__ty_2Erealax_2Ereal(V12x)) = ap(ap(c_2Ereal_2Epow,ap(V3g,inj__ty_2Erealax_2Ereal(V12x))),inj__ty_2Enum_2Enum(V5n)) ) ) ).
tff(lamtp_f2890,type,
f2890: $i > $i ).
tff(lameq_f2890,axiom,
! [V3g: $i] :
( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V13x: tp__ty_2Erealax_2Ereal] : ( ap(f2890(V3g),inj__ty_2Erealax_2Ereal(V13x)) = ap(c_2Etransc_2Eexp,ap(V3g,inj__ty_2Erealax_2Ereal(V13x))) ) ) ).
tff(lamtp_f2891,type,
f2891: $i > $i ).
tff(lameq_f2891,axiom,
! [V3g: $i] :
( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V14x: tp__ty_2Erealax_2Ereal] : ( ap(f2891(V3g),inj__ty_2Erealax_2Ereal(V14x)) = ap(c_2Etransc_2Esin,ap(V3g,inj__ty_2Erealax_2Ereal(V14x))) ) ) ).
tff(lamtp_f2892,type,
f2892: $i > $i ).
tff(lameq_f2892,axiom,
! [V3g: $i] :
( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V15x: tp__ty_2Erealax_2Ereal] : ( ap(f2892(V3g),inj__ty_2Erealax_2Ereal(V15x)) = ap(c_2Etransc_2Ecos,ap(V3g,inj__ty_2Erealax_2Ereal(V15x))) ) ) ).
tff(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,f2883(V0f)),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,f2884(V0f,V3g)),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,f2885(V0f,V3g)),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,f2886(V0f,V3g)),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,f2887(V0f,V3g)),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,f2888(V0f)),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,f2889(V3g,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,f2890(V3g)),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,f2891(V3g)),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,f2892(V3g)),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))) ) ) ) ) ).
tff(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))))) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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))) ) ).
tff(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))))) ) ).
tff(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))))) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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)))) ).
tff(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))) ) ).
tff(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)))) ).
tff(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))) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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))) ) ).
tff(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))) ) ).
tff(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 ) ) ).
tff(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 ) ) ) ).
tff(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 ) ) ).
tff(lamtp_f2893,type,
f2893: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2893,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1u: tp__ty_2Erealax_2Ereal] : ( ap(f2893(V0x),inj__ty_2Erealax_2Ereal(V1u)) = ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(c_2Etransc_2Eexp,inj__ty_2Erealax_2Ereal(V1u))),inj__ty_2Erealax_2Ereal(V0x)) ) ).
tff(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),f2893(V0x))) ) ).
tff(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 ) ).
tff(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))) ) ).
tff(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)))) ) ) ).
tff(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 ) ) ) ).
tff(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))) ).
tff(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)))) ) ) ).
tff(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)))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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)))) ) ) ).
tff(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))) ) ).
tff(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))) ) ).
tff(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)))) ) ).
tff(lamtp_f2894,type,
f2894: ( tp__ty_2Enum_2Enum * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f2894,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V2u: tp__ty_2Erealax_2Ereal] : ( ap(f2894(V0n,V1x),inj__ty_2Erealax_2Ereal(V2u)) = 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))),inj__ty_2Erealax_2Ereal(V2u)))),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V2u)),inj__ty_2Enum_2Enum(V0n))),inj__ty_2Erealax_2Ereal(V1x))) ) ).
tff(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),f2894(V0n,V1x))) ) ).
tff(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))) ) ).
tff(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 ) ) ).
tff(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)))))) ) ) ).
tff(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))) ) ).
tff(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))))) ) ).
tff(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)))) ) ).
tff(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 ) ) ).
tff(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 ) ) ).
tff(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)))) ) ).
tff(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 ) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ).
tff(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))) ).
tff(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))))) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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))) ) ).
tff(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 ) ) ).
tff(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 ) ) ).
tff(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 ) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ).
tff(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)))))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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))) ).
tff(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))))) ).
tff(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))))) ) ).
tff(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)))))) ).
tff(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)))))) ) ).
tff(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)))))) ).
tff(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)))))) ) ).
tff(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))) ) ).
tff(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))) ) ).
tff(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))))) ) ).
tff(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))))) ) ).
tff(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)))) ) ).
tff(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))) ) ).
tff(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))))) ) ).
tff(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)))))) ) ).
tff(lamtp_f2895,type,
f2895: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2895,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] : ( ap(f2895(V0x),inj__ty_2Enum_2Enum(V1n)) = 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)))))),inj__ty_2Enum_2Enum(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)))),inj__ty_2Enum_2Enum(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)))),inj__ty_2Enum_2Enum(V1n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ).
tff(conj_thm_2Etransc_2ESIN__PAIRED,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Eseq_2Esums,f2895(V0x)),ap(c_2Etransc_2Esin,inj__ty_2Erealax_2Ereal(V0x)))) ).
tff(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)))) ) ).
tff(lamtp_f2896,type,
f2896: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2896,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] : ( ap(f2896(V0x),inj__ty_2Enum_2Enum(V1n)) = 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)))))),inj__ty_2Enum_2Enum(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)))),inj__ty_2Enum_2Enum(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)))),inj__ty_2Enum_2Enum(V1n)))) ) ).
tff(conj_thm_2Etransc_2ECOS__PAIRED,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Eseq_2Esums,f2896(V0x)),ap(c_2Etransc_2Ecos,inj__ty_2Erealax_2Ereal(V0x)))) ).
tff(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)))) ).
tff(lamtp_f2897,type,
f2897: $i ).
tff(lameq_f2897,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( ap(f2897,inj__ty_2Erealax_2Ereal(V0x)) = 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))),inj__ty_2Erealax_2Ereal(V0x))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ereal_2Ereal__lte,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)))))),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,inj__ty_2Erealax_2Ereal(V0x))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))) ) ).
tff(conj_thm_2Etransc_2ECOS__ISZERO,axiom,
p(ap(c_2Ebool_2E_3F_21(ty_2Erealax_2Ereal),f2897)) ).
tff(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),f2897))) ).
tff(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),f2897)) ).
tff(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))) ).
tff(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)))))) ) ).
tff(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))) ).
tff(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))))) ).
tff(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)))))) ).
tff(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))) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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))) ) ).
tff(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))) ) ).
tff(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))) ) ).
tff(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))) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(lamtp_f2898,type,
f2898: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2898,axiom,
! [V0y: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] : ( ap(f2898(V0y),inj__ty_2Erealax_2Ereal(V1x)) = 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))),inj__ty_2Erealax_2Ereal(V1x))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(fo__c_2Etransc_2Epi))),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,inj__ty_2Erealax_2Ereal(V1x))),inj__ty_2Erealax_2Ereal(V0y)))) ) ).
tff(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),f2898(V0y))) ) ).
tff(lamtp_f2899,type,
f2899: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2899,axiom,
! [V0y: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] : ( ap(f2899(V0y),inj__ty_2Erealax_2Ereal(V1x)) = 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))))))),inj__ty_2Erealax_2Ereal(V1x))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ereal_2Ereal__lte,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))))))),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(c_2Etransc_2Esin,inj__ty_2Erealax_2Ereal(V1x))),inj__ty_2Erealax_2Ereal(V0y)))) ) ).
tff(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),f2899(V0y))) ) ).
tff(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))))))) ) ) ) ).
tff(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))))))) ) ) ) ).
tff(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)))))))) ) ) ) ) ).
tff(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)))))))) ) ) ) ) ).
tff(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)))) ) ).
tff(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))) ).
tff(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))) ).
tff(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))) ) ).
tff(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)))) ) ).
tff(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))) ) ).
tff(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)))))) ) ) ).
tff(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))))))) ) ) ).
tff(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)))) ) ).
tff(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))) ) ).
tff(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)))) ) ) ).
tff(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 ) ) ) ).
tff(lamtp_f2900,type,
f2900: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2900,axiom,
! [V0y: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] : ( ap(f2900(V0y),inj__ty_2Erealax_2Ereal(V1x)) = 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))))))),inj__ty_2Erealax_2Ereal(V1x))),ap(ap(c_2Ebool_2E_2F_5C,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))))))),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(c_2Etransc_2Etan,inj__ty_2Erealax_2Ereal(V1x))),inj__ty_2Erealax_2Ereal(V0y)))) ) ).
tff(conj_thm_2Etransc_2ETAN__TOTAL,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] : p(ap(c_2Ebool_2E_3F_21(ty_2Erealax_2Ereal),f2900(V0y))) ).
tff(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),f2899(V0y))) ) ).
tff(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),f2898(V0y))) ) ).
tff(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),f2900(V0y))) ) ).
tff(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 ) ) ) ).
tff(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 ) ) ).
tff(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))))))) ) ) ).
tff(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))))))) ) ) ).
tff(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 ) ) ).
tff(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 ) ) ) ).
tff(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 ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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 ) ) ).
tff(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 ) ) ).
tff(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 ) ).
tff(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))))))) ) ).
tff(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 ) ) ).
tff(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))))) ) ) ).
tff(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))))))) ) ) ).
tff(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))))))) ) ) ).
tff(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))) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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))))))) ) ) ).
tff(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))))))) ) ) ).
tff(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))) ) ).
tff(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))) ) ).
tff(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))) ) ).
tff(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))) ) ).
tff(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))) ) ).
tff(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))) ).
tff(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 ) ) ) ) ) ) ).
tff(lamtp_f2901,type,
f2901: ( tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f2901,axiom,
! [V1N: tp__ty_2Enum_2Enum,V0D: $i] :
( mem(V0D,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2901(V1N,V0D),inj__ty_2Enum_2Enum(V2n)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V1N))),ap(ap(c_2Erealax_2Ereal__lt,ap(V0D,inj__ty_2Enum_2Enum(V2n))),ap(V0D,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n))))) ) ) ).
tff(lamtp_f2902,type,
f2902: ( $i * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f2902,axiom,
! [V0D: $i] :
( mem(V0D,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1N: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ( ap(f2902(V0D,V1N),inj__ty_2Enum_2Enum(V3n)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Earithmetic_2E_3E_3D,inj__ty_2Enum_2Enum(V3n)),inj__ty_2Enum_2Enum(V1N))),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(V0D,inj__ty_2Enum_2Enum(V3n))),ap(V0D,inj__ty_2Enum_2Enum(V1N)))) ) ) ).
tff(lamtp_f2903,type,
f2903: $i > $i ).
tff(lameq_f2903,axiom,
! [V0D: $i] :
( mem(V0D,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1N: tp__ty_2Enum_2Enum] : ( ap(f2903(V0D),inj__ty_2Enum_2Enum(V1N)) = ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_21(ty_2Enum_2Enum),f2901(V1N,V0D))),ap(c_2Ebool_2E_21(ty_2Enum_2Enum),f2902(V0D,V1N))) ) ) ).
tff(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),f2903(V0D))) ) ) ).
tff(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))))) ) ) ) ) ) ).
tff(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)))) ) ) ) ) ).
tff(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))))) ) ) ) ) ) ).
tff(lamtp_f2904,type,
f2904: ( $i * $i * $i ) > $i ).
tff(lameq_f2904,axiom,
! [V1p: $i] :
( mem(V1p,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2f: $i] :
( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V0D: $i] :
( mem(V0D,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n: tp__ty_2Enum_2Enum] : ( ap(f2904(V1p,V2f,V0D),inj__ty_2Enum_2Enum(V3n)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V2f,ap(V1p,inj__ty_2Enum_2Enum(V3n)))),ap(ap(c_2Ereal_2Ereal__sub,ap(V0D,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))),ap(V0D,inj__ty_2Enum_2Enum(V3n)))) ) ) ) ) ).
tff(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))),f2904(V1p,V2f,V0D))) ) ) ) ) ).
tff(lamtp_f2905,type,
f2905: ( tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f2905,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V6x: tp__ty_2Erealax_2Ereal] : ( ap(f2905(V0a,V1b),inj__ty_2Erealax_2Ereal(V6x)) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V6x))),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V1b))) ) ).
tff(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,f2905(V0a,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))) ) ) ) ) ) ) ) ).
tff(lamtp_f2906,type,
f2906: ( tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f2906,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum] : ( ap(f2906(V0a,V1b),inj__ty_2Enum_2Enum(V2n)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b)) ) ).
tff(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,f2906(V0a,V1b))) = fo__c_2Enum_2E0 ) ) ).
tff(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,f2906(V0a,V1b))) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) ) ) ).
tff(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))),f2906(V0a,V1b))) ) ).
tff(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 ) ) ) ).
tff(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 ) ) ) ) ) ).
tff(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 ) ) ) ).
tff(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)))) ) ) ).
tff(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))))) ) ) ) ).
tff(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))) ) ) ).
tff(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)))) ) ) ) ).
tff(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 ) ) ) ) ).
tff(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)))) ) ) ).
tff(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))))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ) ) ).
tff(lamtp_f2907,type,
f2907: ( tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f2907,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V3x: tp__ty_2Erealax_2Ereal] : ( ap(f2907(V0a,V1b),inj__ty_2Erealax_2Ereal(V3x)) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V3x))),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V1b))) ) ).
tff(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,f2907(V0a,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))) ) ) ) ) ).
tff(lamtp_f2908,type,
f2908: ( $i * $i ) > $i ).
tff(lameq_f2908,axiom,
! [V1g1: $i] :
( mem(V1g1,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2g2: $i] :
( mem(V2g2,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3x: tp__ty_2Erealax_2Ereal] : ( ap(f2908(V1g1,V2g2),inj__ty_2Erealax_2Ereal(V3x)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__lt,ap(V1g1,inj__ty_2Erealax_2Ereal(V3x))),ap(V2g2,inj__ty_2Erealax_2Ereal(V3x)))),ap(V1g1,inj__ty_2Erealax_2Ereal(V3x))),ap(V2g2,inj__ty_2Erealax_2Ereal(V3x))) ) ) ) ).
tff(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),f2908(V1g1,V2g2))) ) ) ) ) ).
tff(lamtp_f2909,type,
f2909: ( $i * $i ) > $i ).
tff(lameq_f2909,axiom,
! [V0g1: $i] :
( mem(V0g1,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g2: $i] :
( mem(V1g2,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V4x: tp__ty_2Erealax_2Ereal] : ( ap(f2909(V0g1,V1g2),inj__ty_2Erealax_2Ereal(V4x)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__lt,ap(V0g1,inj__ty_2Erealax_2Ereal(V4x))),ap(V1g2,inj__ty_2Erealax_2Ereal(V4x)))),ap(V0g1,inj__ty_2Erealax_2Ereal(V4x))),ap(V1g2,inj__ty_2Erealax_2Ereal(V4x))) ) ) ) ).
tff(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,f2909(V0g1,V1g2)),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))) ) ) ) ) ) ) ).
tff(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 ) ) ) ).
tff(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)))) ) ).
tff(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))))) ) ) ) ).
tff(lamtp_f2910,type,
f2910: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f2910,axiom,
! [V1diff: $i] :
( mem(V1diff,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))
=> ! [V2h: tp__ty_2Erealax_2Ereal,V7m: tp__ty_2Enum_2Enum] : ( ap(f2910(V1diff,V2h),inj__ty_2Enum_2Enum(V7m)) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2E_2F,ap(ap(V1diff,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(V7m))))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V2h)),inj__ty_2Enum_2Enum(V7m))) ) ) ).
tff(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))),f2910(V1diff,V2h))),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))))) ) ) ) ) ) ).
tff(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))),f2910(V1diff,V2h))),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))))) ) ) ) ) ) ).
tff(lamtp_f2911,type,
f2911: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f2911,axiom,
! [V1diff: $i] :
( mem(V1diff,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))
=> ! [V4x: tp__ty_2Erealax_2Ereal,V7m: tp__ty_2Enum_2Enum] : ( ap(f2911(V1diff,V4x),inj__ty_2Enum_2Enum(V7m)) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2E_2F,ap(ap(V1diff,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(V7m))))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Enum_2Enum(V7m))) ) ) ).
tff(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))),f2911(V1diff,V4x))),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))))) ) ) ) ) ) ) ).
tff(lamtp_f2912,type,
f2912: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f2912,axiom,
! [V0diff: $i] :
( mem(V0diff,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Enum_2Enum] : ( ap(f2912(V0diff,V2x),inj__ty_2Enum_2Enum(V3m)) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2E_2F,ap(ap(V0diff,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(V3m))))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Enum_2Enum(V3m))) ) ) ).
tff(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))),f2912(V0diff,V2x))) = 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)))) ) ) ) ).
tff(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))),f2911(V1diff,V4x))),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))))) ) ) ) ) ) ).
tff(lamtp_f2913,type,
f2913: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2913,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Enum_2Enum] : ( ap(f2913(V0x),inj__ty_2Enum_2Enum(V3m)) = ap(ap(c_2Ereal_2E_2F,ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V3m))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2EFACT,inj__ty_2Enum_2Enum(V3m)))) ) ).
tff(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))),f2913(V0x))),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))))) ) ) ) ).
tff(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))),f2913(V0x))),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))))) ) ) ).
tff(lamtp_f2914,type,
f2914: $i > $i ).
tff(lameq_f2914,axiom,
! [V0g: $i] :
( mem(V0g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3x: tp__ty_2Erealax_2Ereal] : ( ap(f2914(V0g),inj__ty_2Erealax_2Ereal(V3x)) = ap(c_2Etransc_2Eln,ap(V0g,inj__ty_2Erealax_2Ereal(V3x))) ) ) ).
tff(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,f2914(V0g)),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))) ) ) ).
tff(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))))) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ) ).
tff(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))))) ) ) ).
tff(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 ) ) ).
tff(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))))) ) ) ).
tff(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)))) ) ).
tff(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))) ) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ).
tff(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))) ) ).
tff(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)))) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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 ) ) ).
tff(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 ) ) ).
tff(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))))))) ) ) ).
tff(lamtp_f2915,type,
f2915: $i > $i ).
tff(lameq_f2915,axiom,
! [V0g: $i] :
( mem(V0g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3x: tp__ty_2Erealax_2Ereal] : ( ap(f2915(V0g),inj__ty_2Erealax_2Ereal(V3x)) = ap(c_2Etransc_2Eexp,ap(V0g,inj__ty_2Erealax_2Ereal(V3x))) ) ) ).
tff(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,f2915(V0g)),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))) ) ) ).
tff(lamtp_f2916,type,
f2916: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2916,axiom,
! [V1y: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] : ( ap(f2916(V1y),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Etransc_2Erpow,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1y)) ) ).
tff(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,f2916(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))) ) ).
%------------------------------------------------------------------------------