ITP001 Axioms: ITP024^5.ax
%------------------------------------------------------------------------------
% File : ITP024^5 : 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 : logroot^2.ax [Gau20]
% : HL4024^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 52 ( 9 unt; 9 typ; 0 def)
% Number of atoms : 1807 ( 37 equ; 0 cnn)
% Maximal formula atoms : 52 ( 34 avg)
% Number of connectives : 2056 ( 0 ~; 0 |; 34 &;1989 @)
% ( 5 <=>; 28 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 9 avg;1989 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 4 ( 4 >; 0 *; 0 +; 0 <<)
% Number of symbols : 41 ( 40 usr; 38 con; 0-2 aty)
% Number of variables : 92 ( 13 ^ 77 !; 2 ?; 92 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Elogroot_2ELOG,type,
c_2Elogroot_2ELOG: $i ).
thf(mem_c_2Elogroot_2ELOG,axiom,
mem @ c_2Elogroot_2ELOG @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Elogroot_2ELOG,type,
fo__c_2Elogroot_2ELOG: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Elogroot_2ELOG,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Elogroot_2ELOG @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Elogroot_2EROOT,type,
c_2Elogroot_2EROOT: $i ).
thf(mem_c_2Elogroot_2EROOT,axiom,
mem @ c_2Elogroot_2EROOT @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Elogroot_2EROOT,type,
fo__c_2Elogroot_2EROOT: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Elogroot_2EROOT,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Elogroot_2EROOT @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Elogroot_2ESQRTd,type,
c_2Elogroot_2ESQRTd: $i ).
thf(mem_c_2Elogroot_2ESQRTd,axiom,
mem @ c_2Elogroot_2ESQRTd @ ( arr @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Elogroot_2EiSQRT0,type,
c_2Elogroot_2EiSQRT0: $i ).
thf(mem_c_2Elogroot_2EiSQRT0,axiom,
mem @ c_2Elogroot_2EiSQRT0 @ ( arr @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Elogroot_2EiSQRT1,type,
c_2Elogroot_2EiSQRT1: $i ).
thf(mem_c_2Elogroot_2EiSQRT1,axiom,
mem @ c_2Elogroot_2EiSQRT1 @ ( arr @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Elogroot_2EiSQRT2,type,
c_2Elogroot_2EiSQRT2: $i ).
thf(mem_c_2Elogroot_2EiSQRT2,axiom,
mem @ c_2Elogroot_2EiSQRT2 @ ( arr @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Elogroot_2EiSQRT3,type,
c_2Elogroot_2EiSQRT3: $i ).
thf(mem_c_2Elogroot_2EiSQRT3,axiom,
mem @ c_2Elogroot_2EiSQRT3 @ ( arr @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(conj_thm_2Elogroot_2ELT__EXP__ISO,axiom,
! [V0e: tp__ty_2Enum_2Enum,V1a: tp__ty_2Enum_2Enum,V2b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0e ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0e ) ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0e ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2ELE__EXP__ISO,axiom,
! [V0e: tp__ty_2Enum_2Enum,V1a: tp__ty_2Enum_2Enum,V2b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0e ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0e ) ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0e ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2EEXP__LT__ISO,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2r: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2r ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V2r ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2r ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2EEXP__LE__ISO,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2r: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2r ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V2r ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2r ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2EROOT__exists,axiom,
! [V0r: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) )
=> ? [V2rt: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V2rt ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2rt ) ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) ) ) ) ) ).
thf(ax_thm_2Elogroot_2EROOT,axiom,
! [V0r: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( inj__ty_2Enum_2Enum @ V0r ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( inj__ty_2Enum_2Enum @ V0r ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2EROOT__UNIQUE,axiom,
! [V0r: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( inj__ty_2Enum_2Enum @ V0r ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= V2p ) ) ).
thf(conj_thm_2Elogroot_2ELOG__exists,axiom,
? [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) )
& ! [V1a: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( ap @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elogroot_2ELOG,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2ELOG__UNIQUE,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= V2p ) ) ).
thf(conj_thm_2Elogroot_2ELOG__ADD1,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1a: tp__ty_2Enum_2Enum,V2b: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2ELOG__BASE,axiom,
! [V0a: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2ELOG__EXP,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1a: tp__ty_2Enum_2Enum,V2b: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2ELOG__1,axiom,
! [V0a: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= fo__c_2Enum_2E0 ) ) ).
thf(conj_thm_2Elogroot_2ELOG__DIV,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2ELOG__ADD,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) )
= V2c ) ) ).
thf(conj_thm_2Elogroot_2ELOG__LE__MONO,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum,V2y: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2ELOG__RWT,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2ELOG__EQ__0,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
= fo__c_2Enum_2E0 )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2ELOG__MULT,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2ELOG__add__digit,axiom,
! [V0b: tp__ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum,V2y: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2EROOT__DIV,axiom,
! [V0r: tp__ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum,V2y: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( inj__ty_2Enum_2Enum @ V0r ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( inj__ty_2Enum_2Enum @ V0r ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V2y ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2EROOT__LE__MONO,axiom,
! [V0r: tp__ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum,V2y: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( inj__ty_2Enum_2Enum @ V0r ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( inj__ty_2Enum_2Enum @ V0r ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2EEXP__MUL,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2ELOG__ROOT,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum,V2r: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2r ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( inj__ty_2Enum_2Enum @ V2r ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) @ ( inj__ty_2Enum_2Enum @ V2r ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2ELOG__MOD,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
=> ( V0n
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Elogroot_2ELOG @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2EROOT__COMPUTE,axiom,
! [V0r: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( inj__ty_2Enum_2Enum @ V0r ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= fo__c_2Enum_2E0 )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( inj__ty_2Enum_2Enum @ V0r ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Enum_2ESUC @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) ) ) @ V2x ) @ ( ap @ c_2Enum_2ESUC @ V2x ) ) ) )
@ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( inj__ty_2Enum_2Enum @ V0r ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0r ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elogroot_2ESQRTd__def,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2ESQRTd @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ) ).
thf(ax_thm_2Elogroot_2EiSQRT0__def,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2EiSQRT0 @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ^ [V1p: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2d: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V2d ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2d ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Epair_2ESND @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) ) )
@ ( ap @ c_2Elogroot_2ESQRTd @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(ax_thm_2Elogroot_2EiSQRT1__def,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2EiSQRT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ^ [V1p: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2d: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V2d ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V2d ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( c_2Epair_2ESND @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) ) )
@ ( ap @ c_2Elogroot_2ESQRTd @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(ax_thm_2Elogroot_2EiSQRT2__def,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2EiSQRT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ^ [V1p: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2d: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3c: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4e: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V4e ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V2d ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V3c ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Enum_2ESUC @ V2d ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V4e ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V3c ) @ V2d ) ) ) )
@ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) ) )
@ ( ap @ c_2Elogroot_2ESQRTd @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(ax_thm_2Elogroot_2EiSQRT3__def,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2EiSQRT3 @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ^ [V1p: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2d: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3c: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4e: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V4e ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V2d ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V3c ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Enum_2ESUC @ V2d ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V4e ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Enum_2ESUC @ V3c ) ) @ V2d ) ) ) )
@ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1p ) ) ) ) )
@ ( ap @ c_2Elogroot_2ESQRTd @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2Enumeral__sqrt,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2ESQRTd @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2ESQRTd @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2ESQRTd @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2ESQRTd @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2EiSQRT3 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2ESQRTd @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2EiSQRT0 @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
& ( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2ESQRTd @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2EiSQRT1 @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
& ( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2ESQRTd @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2EiSQRT2 @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
& ( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2ESQRTd @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2EiSQRT0 @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
& ( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2ESQRTd @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2EiSQRT1 @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
& ( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2ESQRTd @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2EiSQRT2 @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
& ( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2ESQRTd @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Elogroot_2EiSQRT3 @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).
thf(conj_thm_2Elogroot_2Enumeral__root2,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Elogroot_2EROOT @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Elogroot_2ESQRTd @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
%------------------------------------------------------------------------------