ITP001 Axioms: ITP126^5.ax
%------------------------------------------------------------------------------
% File : ITP126^5 : 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 : rat^2.ax [Gau20]
% : HL4126^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 353 ( 116 unt; 51 typ; 0 def)
% Number of atoms : 4191 ( 316 equ; 0 cnn)
% Maximal formula atoms : 64 ( 11 avg)
% Number of connectives : 6605 ( 46 ~; 14 |; 76 &;6277 @)
% ( 116 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg;6277 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 27 ( 27 >; 0 *; 0 +; 0 <<)
% Number of symbols : 121 ( 118 usr; 100 con; 0-2 aty)
% Number of variables : 519 ( 13 ^ 501 !; 5 ?; 519 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Erat_2Erat,type,
ty_2Erat_2Erat: del ).
thf(stp_ty_2Erat_2Erat,type,
tp__ty_2Erat_2Erat: $tType ).
thf(stp_inj_ty_2Erat_2Erat,type,
inj__ty_2Erat_2Erat: tp__ty_2Erat_2Erat > $i ).
thf(stp_surj_ty_2Erat_2Erat,type,
surj__ty_2Erat_2Erat: $i > tp__ty_2Erat_2Erat ).
thf(stp_inj_surj_ty_2Erat_2Erat,axiom,
! [X: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( inj__ty_2Erat_2Erat @ X ) )
= X ) ).
thf(stp_inj_mem_ty_2Erat_2Erat,axiom,
! [X: tp__ty_2Erat_2Erat] : ( mem @ ( inj__ty_2Erat_2Erat @ X ) @ ty_2Erat_2Erat ) ).
thf(stp_iso_mem_ty_2Erat_2Erat,axiom,
! [X: $i] :
( ( mem @ X @ ty_2Erat_2Erat )
=> ( X
= ( inj__ty_2Erat_2Erat @ ( surj__ty_2Erat_2Erat @ X ) ) ) ) ).
thf(tp_c_2Erat_2ERATD,type,
c_2Erat_2ERATD: $i ).
thf(mem_c_2Erat_2ERATD,axiom,
mem @ c_2Erat_2ERATD @ ( arr @ ty_2Erat_2Erat @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Erat_2ERATD,type,
fo__c_2Erat_2ERATD: tp__ty_2Erat_2Erat > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Erat_2ERATD,axiom,
! [X0: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Erat_2ERATD @ X0 ) )
= ( ap @ c_2Erat_2ERATD @ ( inj__ty_2Erat_2Erat @ X0 ) ) ) ).
thf(tp_c_2Erat_2ERATN,type,
c_2Erat_2ERATN: $i ).
thf(mem_c_2Erat_2ERATN,axiom,
mem @ c_2Erat_2ERATN @ ( arr @ ty_2Erat_2Erat @ ty_2Einteger_2Eint ) ).
thf(stp_fo_c_2Erat_2ERATN,type,
fo__c_2Erat_2ERATN: tp__ty_2Erat_2Erat > tp__ty_2Einteger_2Eint ).
thf(stp_eq_fo_c_2Erat_2ERATN,axiom,
! [X0: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Einteger_2Eint @ ( fo__c_2Erat_2ERATN @ X0 ) )
= ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ X0 ) ) ) ).
thf(tp_c_2Erat_2Eabs__rat,type,
c_2Erat_2Eabs__rat: $i ).
thf(mem_c_2Erat_2Eabs__rat,axiom,
mem @ c_2Erat_2Eabs__rat @ ( arr @ ty_2Efrac_2Efrac @ ty_2Erat_2Erat ) ).
thf(stp_fo_c_2Erat_2Eabs__rat,type,
fo__c_2Erat_2Eabs__rat: tp__ty_2Efrac_2Efrac > tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Eabs__rat,axiom,
! [X0: tp__ty_2Efrac_2Efrac] :
( ( inj__ty_2Erat_2Erat @ ( fo__c_2Erat_2Eabs__rat @ X0 ) )
= ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ X0 ) ) ) ).
thf(tp_c_2Erat_2Eabs__rat__CLASS,type,
c_2Erat_2Eabs__rat__CLASS: $i ).
thf(mem_c_2Erat_2Eabs__rat__CLASS,axiom,
mem @ c_2Erat_2Eabs__rat__CLASS @ ( arr @ ( arr @ ty_2Efrac_2Efrac @ bool ) @ ty_2Erat_2Erat ) ).
thf(tp_c_2Erat_2Erat__0,type,
c_2Erat_2Erat__0: $i ).
thf(mem_c_2Erat_2Erat__0,axiom,
mem @ c_2Erat_2Erat__0 @ ty_2Erat_2Erat ).
thf(stp_fo_c_2Erat_2Erat__0,type,
fo__c_2Erat_2Erat__0: tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__0,axiom,
( ( inj__ty_2Erat_2Erat @ fo__c_2Erat_2Erat__0 )
= c_2Erat_2Erat__0 ) ).
thf(tp_c_2Erat_2Erat__1,type,
c_2Erat_2Erat__1: $i ).
thf(mem_c_2Erat_2Erat__1,axiom,
mem @ c_2Erat_2Erat__1 @ ty_2Erat_2Erat ).
thf(stp_fo_c_2Erat_2Erat__1,type,
fo__c_2Erat_2Erat__1: tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__1,axiom,
( ( inj__ty_2Erat_2Erat @ fo__c_2Erat_2Erat__1 )
= c_2Erat_2Erat__1 ) ).
thf(tp_c_2Erat_2Erat__add,type,
c_2Erat_2Erat__add: $i ).
thf(mem_c_2Erat_2Erat__add,axiom,
mem @ c_2Erat_2Erat__add @ ( arr @ ty_2Erat_2Erat @ ( arr @ ty_2Erat_2Erat @ ty_2Erat_2Erat ) ) ).
thf(stp_fo_c_2Erat_2Erat__add,type,
fo__c_2Erat_2Erat__add: tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__add,axiom,
! [X0: tp__ty_2Erat_2Erat,X1: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Erat_2Erat @ ( fo__c_2Erat_2Erat__add @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ X0 ) ) @ ( inj__ty_2Erat_2Erat @ X1 ) ) ) ).
thf(tp_c_2Erat_2Erat__ainv,type,
c_2Erat_2Erat__ainv: $i ).
thf(mem_c_2Erat_2Erat__ainv,axiom,
mem @ c_2Erat_2Erat__ainv @ ( arr @ ty_2Erat_2Erat @ ty_2Erat_2Erat ) ).
thf(stp_fo_c_2Erat_2Erat__ainv,type,
fo__c_2Erat_2Erat__ainv: tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__ainv,axiom,
! [X0: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Erat_2Erat @ ( fo__c_2Erat_2Erat__ainv @ X0 ) )
= ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ X0 ) ) ) ).
thf(tp_c_2Erat_2Erat__cons,type,
c_2Erat_2Erat__cons: $i ).
thf(mem_c_2Erat_2Erat__cons,axiom,
mem @ c_2Erat_2Erat__cons @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ty_2Einteger_2Eint @ ty_2Erat_2Erat ) ) ).
thf(stp_fo_c_2Erat_2Erat__cons,type,
fo__c_2Erat_2Erat__cons: tp__ty_2Einteger_2Eint > tp__ty_2Einteger_2Eint > tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__cons,axiom,
! [X0: tp__ty_2Einteger_2Eint,X1: tp__ty_2Einteger_2Eint] :
( ( inj__ty_2Erat_2Erat @ ( fo__c_2Erat_2Erat__cons @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Erat_2Erat__cons @ ( inj__ty_2Einteger_2Eint @ X0 ) ) @ ( inj__ty_2Einteger_2Eint @ X1 ) ) ) ).
thf(tp_c_2Erat_2Erat__div,type,
c_2Erat_2Erat__div: $i ).
thf(mem_c_2Erat_2Erat__div,axiom,
mem @ c_2Erat_2Erat__div @ ( arr @ ty_2Erat_2Erat @ ( arr @ ty_2Erat_2Erat @ ty_2Erat_2Erat ) ) ).
thf(stp_fo_c_2Erat_2Erat__div,type,
fo__c_2Erat_2Erat__div: tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__div,axiom,
! [X0: tp__ty_2Erat_2Erat,X1: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Erat_2Erat @ ( fo__c_2Erat_2Erat__div @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ X0 ) ) @ ( inj__ty_2Erat_2Erat @ X1 ) ) ) ).
thf(tp_c_2Erat_2Erat__dnm,type,
c_2Erat_2Erat__dnm: $i ).
thf(mem_c_2Erat_2Erat__dnm,axiom,
mem @ c_2Erat_2Erat__dnm @ ( arr @ ty_2Erat_2Erat @ ty_2Einteger_2Eint ) ).
thf(stp_fo_c_2Erat_2Erat__dnm,type,
fo__c_2Erat_2Erat__dnm: tp__ty_2Erat_2Erat > tp__ty_2Einteger_2Eint ).
thf(stp_eq_fo_c_2Erat_2Erat__dnm,axiom,
! [X0: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Einteger_2Eint @ ( fo__c_2Erat_2Erat__dnm @ X0 ) )
= ( ap @ c_2Erat_2Erat__dnm @ ( inj__ty_2Erat_2Erat @ X0 ) ) ) ).
thf(tp_c_2Erat_2Erat__equiv,type,
c_2Erat_2Erat__equiv: $i ).
thf(mem_c_2Erat_2Erat__equiv,axiom,
mem @ c_2Erat_2Erat__equiv @ ( arr @ ty_2Efrac_2Efrac @ ( arr @ ty_2Efrac_2Efrac @ bool ) ) ).
thf(tp_c_2Erat_2Erat__geq,type,
c_2Erat_2Erat__geq: $i ).
thf(mem_c_2Erat_2Erat__geq,axiom,
mem @ c_2Erat_2Erat__geq @ ( arr @ ty_2Erat_2Erat @ ( arr @ ty_2Erat_2Erat @ bool ) ) ).
thf(tp_c_2Erat_2Erat__gre,type,
c_2Erat_2Erat__gre: $i ).
thf(mem_c_2Erat_2Erat__gre,axiom,
mem @ c_2Erat_2Erat__gre @ ( arr @ ty_2Erat_2Erat @ ( arr @ ty_2Erat_2Erat @ bool ) ) ).
thf(tp_c_2Erat_2Erat__leq,type,
c_2Erat_2Erat__leq: $i ).
thf(mem_c_2Erat_2Erat__leq,axiom,
mem @ c_2Erat_2Erat__leq @ ( arr @ ty_2Erat_2Erat @ ( arr @ ty_2Erat_2Erat @ bool ) ) ).
thf(tp_c_2Erat_2Erat__les,type,
c_2Erat_2Erat__les: $i ).
thf(mem_c_2Erat_2Erat__les,axiom,
mem @ c_2Erat_2Erat__les @ ( arr @ ty_2Erat_2Erat @ ( arr @ ty_2Erat_2Erat @ bool ) ) ).
thf(tp_c_2Erat_2Erat__max,type,
c_2Erat_2Erat__max: $i ).
thf(mem_c_2Erat_2Erat__max,axiom,
mem @ c_2Erat_2Erat__max @ ( arr @ ty_2Erat_2Erat @ ( arr @ ty_2Erat_2Erat @ ty_2Erat_2Erat ) ) ).
thf(stp_fo_c_2Erat_2Erat__max,type,
fo__c_2Erat_2Erat__max: tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__max,axiom,
! [X0: tp__ty_2Erat_2Erat,X1: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Erat_2Erat @ ( fo__c_2Erat_2Erat__max @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Erat_2Erat__max @ ( inj__ty_2Erat_2Erat @ X0 ) ) @ ( inj__ty_2Erat_2Erat @ X1 ) ) ) ).
thf(tp_c_2Erat_2Erat__min,type,
c_2Erat_2Erat__min: $i ).
thf(mem_c_2Erat_2Erat__min,axiom,
mem @ c_2Erat_2Erat__min @ ( arr @ ty_2Erat_2Erat @ ( arr @ ty_2Erat_2Erat @ ty_2Erat_2Erat ) ) ).
thf(stp_fo_c_2Erat_2Erat__min,type,
fo__c_2Erat_2Erat__min: tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__min,axiom,
! [X0: tp__ty_2Erat_2Erat,X1: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Erat_2Erat @ ( fo__c_2Erat_2Erat__min @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Erat_2Erat__min @ ( inj__ty_2Erat_2Erat @ X0 ) ) @ ( inj__ty_2Erat_2Erat @ X1 ) ) ) ).
thf(tp_c_2Erat_2Erat__minv,type,
c_2Erat_2Erat__minv: $i ).
thf(mem_c_2Erat_2Erat__minv,axiom,
mem @ c_2Erat_2Erat__minv @ ( arr @ ty_2Erat_2Erat @ ty_2Erat_2Erat ) ).
thf(stp_fo_c_2Erat_2Erat__minv,type,
fo__c_2Erat_2Erat__minv: tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__minv,axiom,
! [X0: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Erat_2Erat @ ( fo__c_2Erat_2Erat__minv @ X0 ) )
= ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ X0 ) ) ) ).
thf(tp_c_2Erat_2Erat__mul,type,
c_2Erat_2Erat__mul: $i ).
thf(mem_c_2Erat_2Erat__mul,axiom,
mem @ c_2Erat_2Erat__mul @ ( arr @ ty_2Erat_2Erat @ ( arr @ ty_2Erat_2Erat @ ty_2Erat_2Erat ) ) ).
thf(stp_fo_c_2Erat_2Erat__mul,type,
fo__c_2Erat_2Erat__mul: tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__mul,axiom,
! [X0: tp__ty_2Erat_2Erat,X1: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Erat_2Erat @ ( fo__c_2Erat_2Erat__mul @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ X0 ) ) @ ( inj__ty_2Erat_2Erat @ X1 ) ) ) ).
thf(tp_c_2Erat_2Erat__nmr,type,
c_2Erat_2Erat__nmr: $i ).
thf(mem_c_2Erat_2Erat__nmr,axiom,
mem @ c_2Erat_2Erat__nmr @ ( arr @ ty_2Erat_2Erat @ ty_2Einteger_2Eint ) ).
thf(stp_fo_c_2Erat_2Erat__nmr,type,
fo__c_2Erat_2Erat__nmr: tp__ty_2Erat_2Erat > tp__ty_2Einteger_2Eint ).
thf(stp_eq_fo_c_2Erat_2Erat__nmr,axiom,
! [X0: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Einteger_2Eint @ ( fo__c_2Erat_2Erat__nmr @ X0 ) )
= ( ap @ c_2Erat_2Erat__nmr @ ( inj__ty_2Erat_2Erat @ X0 ) ) ) ).
thf(tp_c_2Erat_2Erat__of__int,type,
c_2Erat_2Erat__of__int: $i ).
thf(mem_c_2Erat_2Erat__of__int,axiom,
mem @ c_2Erat_2Erat__of__int @ ( arr @ ty_2Einteger_2Eint @ ty_2Erat_2Erat ) ).
thf(stp_fo_c_2Erat_2Erat__of__int,type,
fo__c_2Erat_2Erat__of__int: tp__ty_2Einteger_2Eint > tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__of__int,axiom,
! [X0: tp__ty_2Einteger_2Eint] :
( ( inj__ty_2Erat_2Erat @ ( fo__c_2Erat_2Erat__of__int @ X0 ) )
= ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ X0 ) ) ) ).
thf(tp_c_2Erat_2Erat__of__num,type,
c_2Erat_2Erat__of__num: $i ).
thf(mem_c_2Erat_2Erat__of__num,axiom,
mem @ c_2Erat_2Erat__of__num @ ( arr @ ty_2Enum_2Enum @ ty_2Erat_2Erat ) ).
thf(stp_fo_c_2Erat_2Erat__of__num,type,
fo__c_2Erat_2Erat__of__num: tp__ty_2Enum_2Enum > tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__of__num,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Erat_2Erat @ ( fo__c_2Erat_2Erat__of__num @ X0 ) )
= ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Erat_2Erat__sgn,type,
c_2Erat_2Erat__sgn: $i ).
thf(mem_c_2Erat_2Erat__sgn,axiom,
mem @ c_2Erat_2Erat__sgn @ ( arr @ ty_2Erat_2Erat @ ty_2Einteger_2Eint ) ).
thf(stp_fo_c_2Erat_2Erat__sgn,type,
fo__c_2Erat_2Erat__sgn: tp__ty_2Erat_2Erat > tp__ty_2Einteger_2Eint ).
thf(stp_eq_fo_c_2Erat_2Erat__sgn,axiom,
! [X0: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Einteger_2Eint @ ( fo__c_2Erat_2Erat__sgn @ X0 ) )
= ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ X0 ) ) ) ).
thf(tp_c_2Erat_2Erat__sub,type,
c_2Erat_2Erat__sub: $i ).
thf(mem_c_2Erat_2Erat__sub,axiom,
mem @ c_2Erat_2Erat__sub @ ( arr @ ty_2Erat_2Erat @ ( arr @ ty_2Erat_2Erat @ ty_2Erat_2Erat ) ) ).
thf(stp_fo_c_2Erat_2Erat__sub,type,
fo__c_2Erat_2Erat__sub: tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat > tp__ty_2Erat_2Erat ).
thf(stp_eq_fo_c_2Erat_2Erat__sub,axiom,
! [X0: tp__ty_2Erat_2Erat,X1: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Erat_2Erat @ ( fo__c_2Erat_2Erat__sub @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ X0 ) ) @ ( inj__ty_2Erat_2Erat @ X1 ) ) ) ).
thf(tp_c_2Erat_2Erep__rat,type,
c_2Erat_2Erep__rat: $i ).
thf(mem_c_2Erat_2Erep__rat,axiom,
mem @ c_2Erat_2Erep__rat @ ( arr @ ty_2Erat_2Erat @ ty_2Efrac_2Efrac ) ).
thf(stp_fo_c_2Erat_2Erep__rat,type,
fo__c_2Erat_2Erep__rat: tp__ty_2Erat_2Erat > tp__ty_2Efrac_2Efrac ).
thf(stp_eq_fo_c_2Erat_2Erep__rat,axiom,
! [X0: tp__ty_2Erat_2Erat] :
( ( inj__ty_2Efrac_2Efrac @ ( fo__c_2Erat_2Erep__rat @ X0 ) )
= ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ X0 ) ) ) ).
thf(tp_c_2Erat_2Erep__rat__CLASS,type,
c_2Erat_2Erep__rat__CLASS: $i ).
thf(mem_c_2Erat_2Erep__rat__CLASS,axiom,
mem @ c_2Erat_2Erep__rat__CLASS @ ( arr @ ty_2Erat_2Erat @ ( arr @ ty_2Efrac_2Efrac @ bool ) ) ).
thf(ax_thm_2Erat_2Erat__equiv__def,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac,V1f2: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) )
<=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Efrac_2Efrac__dnm @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) @ ( ap @ c_2Efrac_2Efrac__dnm @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQUIV__REF,axiom,
! [V0a: tp__ty_2Efrac_2Efrac] : ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQUIV__SYM,axiom,
! [V0a: tp__ty_2Efrac_2Efrac,V1b: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) @ ( inj__ty_2Efrac_2Efrac @ V1b ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V1b ) ) @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQUIV__NMR__Z__IFF,axiom,
! [V0a: tp__ty_2Efrac_2Efrac,V1b: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) @ ( inj__ty_2Efrac_2Efrac @ V1b ) ) )
=> ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1b ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQUIV__NMR__GTZ__IFF,axiom,
! [V0a: tp__ty_2Efrac_2Efrac,V1b: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) @ ( inj__ty_2Efrac_2Efrac @ V1b ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__gt @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__gt @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1b ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQUIV__NMR__LTZ__IFF,axiom,
! [V0a: tp__ty_2Efrac_2Efrac,V1b: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) @ ( inj__ty_2Efrac_2Efrac @ V1b ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1b ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__NMR__Z__IFF__EQUIV,axiom,
! [V0a: tp__ty_2Efrac_2Efrac,V1b: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) @ ( inj__ty_2Efrac_2Efrac @ V1b ) ) )
<=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1b ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQUIV__TRANS,axiom,
! [V0a: tp__ty_2Efrac_2Efrac,V1b: tp__ty_2Efrac_2Efrac,V2c: tp__ty_2Efrac_2Efrac] :
( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) @ ( inj__ty_2Efrac_2Efrac @ V1b ) ) )
& ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V1b ) ) @ ( inj__ty_2Efrac_2Efrac @ V2c ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) @ ( inj__ty_2Efrac_2Efrac @ V2c ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQUIV,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac,V1f2: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) )
<=> ( ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) )
= ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQUIV__ALT,axiom,
! [V0a: tp__ty_2Efrac_2Efrac] :
( ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0a ) )
= ( lam @ ty_2Efrac_2Efrac
@ ^ [V1x: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Einteger_2Eint )
@ ( lam @ ty_2Einteger_2Eint
@ ^ [V2b: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Einteger_2Eint )
@ ( lam @ ty_2Einteger_2Eint
@ ^ [V3c: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2b ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V3c ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Efrac_2Efrac ) @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( inj__ty_2Efrac_2Efrac @ V0a ) ) @ ( ap @ c_2Efrac_2Eabs__frac @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) @ V2b ) @ V2b ) ) ) ) @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ V1x ) @ ( ap @ c_2Efrac_2Eabs__frac @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) @ V3c ) @ V3c ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__TY__DEF,axiom,
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ty_2Erat_2Erat @ ( arr @ ty_2Efrac_2Efrac @ bool ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ty_2Efrac_2Efrac @ bool ) @ ty_2Erat_2Erat )
@ ( lam @ ( arr @ ty_2Efrac_2Efrac @ bool )
@ ^ [V1c: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Efrac_2Efrac )
@ ( lam @ ty_2Efrac_2Efrac
@ ^ [V2r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ V2r ) @ V2r ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Efrac_2Efrac @ bool ) ) @ V1c ) @ ( ap @ c_2Erat_2Erat__equiv @ V2r ) ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Erat_2Erat__bijections,axiom,
( ! [V0a: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat__CLASS @ ( ap @ c_2Erat_2Erep__rat__CLASS @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) )
= V0a )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ty_2Efrac_2Efrac @ bool ) )
=> ( ( p
@ ( ap
@ ( lam @ ( arr @ ty_2Efrac_2Efrac @ bool )
@ ^ [V2c: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Efrac_2Efrac )
@ ( lam @ ty_2Efrac_2Efrac
@ ^ [V3r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ V3r ) @ V3r ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Efrac_2Efrac @ bool ) ) @ V2c ) @ ( ap @ c_2Erat_2Erat__equiv @ V3r ) ) ) ) ) )
@ V1r ) )
<=> ( ( ap @ c_2Erat_2Erep__rat__CLASS @ ( ap @ c_2Erat_2Eabs__rat__CLASS @ V1r ) )
= V1r ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__ABS__REP__CLASS,axiom,
( ! [V0a: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat__CLASS @ ( ap @ c_2Erat_2Erep__rat__CLASS @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) )
= V0a )
& ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Efrac_2Efrac @ bool ) )
=> ( ? [V2r: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V2r ) ) @ ( inj__ty_2Efrac_2Efrac @ V2r ) ) )
& ( V1c
= ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V2r ) ) ) )
<=> ( ( ap @ c_2Erat_2Erep__rat__CLASS @ ( ap @ c_2Erat_2Eabs__rat__CLASS @ V1c ) )
= V1c ) ) ) ) ).
thf(ax_thm_2Erat_2Erep__rat__def,axiom,
! [V0a: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Efrac_2Efrac @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0a ) ) )
= ( surj__ty_2Efrac_2Efrac @ ( ap @ ( c_2Emin_2E_40 @ ty_2Efrac_2Efrac ) @ ( ap @ c_2Erat_2Erep__rat__CLASS @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) ) ) ).
thf(ax_thm_2Erat_2Eabs__rat__def,axiom,
! [V0r: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0r ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat__CLASS @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0r ) ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__QUOTIENT,axiom,
p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ ty_2Efrac_2Efrac @ ty_2Erat_2Erat ) @ c_2Erat_2Erat__equiv ) @ c_2Erat_2Eabs__rat ) @ c_2Erat_2Erep__rat ) ).
thf(conj_thm_2Erat_2Erat__def,axiom,
p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ ty_2Efrac_2Efrac @ ty_2Erat_2Erat ) @ c_2Erat_2Erat__equiv ) @ c_2Erat_2Eabs__rat ) @ c_2Erat_2Erep__rat ) ).
thf(conj_thm_2Erat_2Erat__type__thm,axiom,
( ! [V0a: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) )
= V0a )
& ! [V1r: tp__ty_2Efrac_2Efrac,V2s: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V1r ) ) @ ( inj__ty_2Efrac_2Efrac @ V2s ) ) )
<=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1r ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__equiv__reps,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
<=> ( V0r1 = V1r2 ) ) ).
thf(conj_thm_2Erat_2Erat__equiv__rep__abs,axiom,
! [V0f: tp__ty_2Efrac_2Efrac] : ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f ) ) ) ) @ ( inj__ty_2Efrac_2Efrac @ V0f ) ) ) ).
thf(ax_thm_2Erat_2Erat__nmr__def,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__nmr @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__dnm__def,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__dnm @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__dnm @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__sgn__def,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__sgn @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__0__def,axiom,
( fo__c_2Erat_2Erat__0
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ fo__c_2Efrac_2Efrac__0 ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__1__def,axiom,
( fo__c_2Erat_2Erat__1
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ fo__c_2Efrac_2Efrac__1 ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__ainv__def,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Efrac_2Efrac__ainv @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__minv__def,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Efrac_2Efrac__minv @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__add__def,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__add @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__sub__def,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__sub @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__mul__def,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__div__def,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__div @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__les__def,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
<=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__gre__def,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__gre @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__leq__def,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
| ( V0r1 = V1r2 ) ) ) ).
thf(ax_thm_2Erat_2Erat__geq__def,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__geq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__cons__def,axiom,
! [V0nmr: tp__ty_2Einteger_2Eint,V1dnm: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__cons @ ( inj__ty_2Einteger_2Eint @ V0nmr ) ) @ ( inj__ty_2Einteger_2Eint @ V1dnm ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Efrac_2Eabs__frac @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2EintExtension_2ESGN @ ( inj__ty_2Einteger_2Eint @ V0nmr ) ) ) @ ( ap @ c_2EintExtension_2ESGN @ ( inj__ty_2Einteger_2Eint @ V1dnm ) ) ) ) @ ( ap @ c_2Einteger_2EABS @ ( inj__ty_2Einteger_2Eint @ V0nmr ) ) ) ) @ ( ap @ c_2Einteger_2EABS @ ( inj__ty_2Einteger_2Eint @ V1dnm ) ) ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__of__num__primitive__def,axiom,
( c_2Erat_2Erat__of__num
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ ty_2Enum_2Enum @ ty_2Erat_2Erat )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) )
@ ^ [V0R: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Erelation_2EWF @ ty_2Enum_2Enum ) @ V0R ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ V0R @ ( ap @ c_2Enum_2ESUC @ V1n ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ V1n ) ) ) ) ) ) ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ty_2Erat_2Erat )
@ ^ [V2rat__of__num: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V3a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2Enum__CASE @ ty_2Erat_2Erat ) @ V3a ) @ ( ap @ ( c_2Ecombin_2EI @ ty_2Erat_2Erat ) @ ( inj__ty_2Erat_2Erat @ fo__c_2Erat_2Erat__0 ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4v: $i] :
( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2Enum__CASE @ ty_2Erat_2Erat ) @ V4v ) @ ( ap @ ( c_2Ecombin_2EI @ ty_2Erat_2Erat ) @ ( inj__ty_2Erat_2Erat @ fo__c_2Erat_2Erat__1 ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( c_2Ecombin_2EI @ ty_2Erat_2Erat ) @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ V2rat__of__num @ ( ap @ c_2Enum_2ESUC @ V5n ) ) ) @ ( inj__ty_2Erat_2Erat @ fo__c_2Erat_2Erat__1 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__of__num__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
& ( p @ ( ap @ V0P @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ V0P @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) )
=> ! [V2v: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2v ) ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__of__num__def,axiom,
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= fo__c_2Erat_2Erat__0 )
& ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= fo__c_2Erat_2Erat__1 )
& ! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( inj__ty_2Erat_2Erat @ fo__c_2Erat_2Erat__1 ) ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__of__num__def__compute,axiom,
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= fo__c_2Erat_2Erat__0 )
& ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= fo__c_2Erat_2Erat__1 )
& ! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) @ ( inj__ty_2Erat_2Erat @ fo__c_2Erat_2Erat__1 ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) @ ( inj__ty_2Erat_2Erat @ fo__c_2Erat_2Erat__1 ) ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__0,axiom,
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ fo__c_2Efrac_2Efrac__0 ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__1,axiom,
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ fo__c_2Efrac_2Efrac__1 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Erat_2Erep__rat @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) )
= V0r ) ).
thf(conj_thm_2Erat_2ERAT__ABS__EQUIV,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac,V1f2: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQ,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac,V1f2: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) )
<=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Efrac_2Efrac__dnm @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) @ ( ap @ c_2Efrac_2Efrac__dnm @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQ__ALT,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( V0r1 = V1r2 )
<=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Erat_2Erat__nmr @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ c_2Erat_2Erat__dnm @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Erat_2Erat__nmr @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( ap @ c_2Erat_2Erat__dnm @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__NMREQ0__CONG,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__NMRLT0__CONG,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ c_2Efrac_2Efrac__nmr @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__NMRGT0__CONG,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__gt @ ( ap @ c_2Efrac_2Efrac__nmr @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__gt @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__CONG,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__sgn @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__sgn @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__AINV__CONG,axiom,
! [V0x: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Efrac_2Efrac__ainv @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Efrac_2Efrac__ainv @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) ) ) ).
thf(conj_thm_2Erat_2EFRAC__MINV__EQUIV,axiom,
! [V0y: tp__ty_2Efrac_2Efrac,V1x: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0y ) ) )
!= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V1x ) ) @ ( inj__ty_2Efrac_2Efrac @ V0y ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( ap @ c_2Efrac_2Efrac__minv @ ( inj__ty_2Efrac_2Efrac @ V1x ) ) ) @ ( ap @ c_2Efrac_2Efrac__minv @ ( inj__ty_2Efrac_2Efrac @ V0y ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MINV__CONG,axiom,
! [V0x: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) )
!= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Efrac_2Efrac__minv @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Efrac_2Efrac__minv @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2EFRAC__ADD__EQUIV1,axiom,
! [V0x: tp__ty_2Efrac_2Efrac,V1x_27: tp__ty_2Efrac_2Efrac,V2y: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1x_27 ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( ap @ ( ap @ c_2Efrac_2Efrac__add @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V2y ) ) ) @ ( ap @ ( ap @ c_2Efrac_2Efrac__add @ ( inj__ty_2Efrac_2Efrac @ V1x_27 ) ) @ ( inj__ty_2Efrac_2Efrac @ V2y ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__ADD__CONG1,axiom,
! [V0x: tp__ty_2Efrac_2Efrac,V1y: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__add @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__add @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__ADD__CONG2,axiom,
! [V0x: tp__ty_2Efrac_2Efrac,V1y: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__add @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__add @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__ADD__CONG,axiom,
( ! [V0x: tp__ty_2Efrac_2Efrac,V1y: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__add @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__add @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) )
& ! [V2x: tp__ty_2Efrac_2Efrac,V3y: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__add @ ( inj__ty_2Efrac_2Efrac @ V2x ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V3y ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__add @ ( inj__ty_2Efrac_2Efrac @ V2x ) ) @ ( inj__ty_2Efrac_2Efrac @ V3y ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2EFRAC__MUL__EQUIV1,axiom,
! [V0x: tp__ty_2Efrac_2Efrac,V1x_27: tp__ty_2Efrac_2Efrac,V2y: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1x_27 ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V2y ) ) ) @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( inj__ty_2Efrac_2Efrac @ V1x_27 ) ) @ ( inj__ty_2Efrac_2Efrac @ V2y ) ) ) ) ) ).
thf(conj_thm_2Erat_2EFRAC__MUL__EQUIV2,axiom,
! [V0x: tp__ty_2Efrac_2Efrac,V1x_27: tp__ty_2Efrac_2Efrac,V2y: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1x_27 ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__equiv @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( inj__ty_2Efrac_2Efrac @ V2y ) ) @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( inj__ty_2Efrac_2Efrac @ V2y ) ) @ ( inj__ty_2Efrac_2Efrac @ V1x_27 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__CONG1,axiom,
! [V0x: tp__ty_2Efrac_2Efrac,V1y: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__CONG2,axiom,
! [V0x: tp__ty_2Efrac_2Efrac,V1y: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__CONG,axiom,
( ! [V0x: tp__ty_2Efrac_2Efrac,V1y: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) )
& ! [V2x: tp__ty_2Efrac_2Efrac,V3y: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( inj__ty_2Efrac_2Efrac @ V2x ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V3y ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( inj__ty_2Efrac_2Efrac @ V2x ) ) @ ( inj__ty_2Efrac_2Efrac @ V3y ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SUB__CONG1,axiom,
! [V0x: tp__ty_2Efrac_2Efrac,V1y: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__sub @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__sub @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SUB__CONG2,axiom,
! [V0x: tp__ty_2Efrac_2Efrac,V1y: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__sub @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__sub @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SUB__CONG,axiom,
( ! [V0x: tp__ty_2Efrac_2Efrac,V1y: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__sub @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__sub @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) )
& ! [V2x: tp__ty_2Efrac_2Efrac,V3y: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__sub @ ( inj__ty_2Efrac_2Efrac @ V2x ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V3y ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__sub @ ( inj__ty_2Efrac_2Efrac @ V2x ) ) @ ( inj__ty_2Efrac_2Efrac @ V3y ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIV__CONG1,axiom,
! [V0x: tp__ty_2Efrac_2Efrac,V1y: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) )
!= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__div @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__div @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIV__CONG2,axiom,
! [V0x: tp__ty_2Efrac_2Efrac,V1y: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) )
!= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__div @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__div @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIV__CONG,axiom,
( ! [V0x: tp__ty_2Efrac_2Efrac,V1y: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) )
!= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__div @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__div @ ( inj__ty_2Efrac_2Efrac @ V0x ) ) @ ( inj__ty_2Efrac_2Efrac @ V1y ) ) ) ) ) )
& ! [V2x: tp__ty_2Efrac_2Efrac,V3y: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V3y ) ) )
!= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__div @ ( inj__ty_2Efrac_2Efrac @ V2x ) ) @ ( ap @ c_2Erat_2Erep__rat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V3y ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__div @ ( inj__ty_2Efrac_2Efrac @ V2x ) ) @ ( inj__ty_2Efrac_2Efrac @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__NMRDNM__EQ,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Efrac_2Eabs__frac @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Efrac_2Efrac__dnm @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
<=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__dnm @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__AINV__CALCULATE,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Efrac_2Efrac__ainv @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MINV__CALCULATE,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
!= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__minv @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Efrac_2Efrac__minv @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__ADD__CALCULATE,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac,V1f2: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__add @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SUB__CALCULATE,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac,V1f2: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__sub @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__CALCULATE,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac,V1f2: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__mul @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIV__CALCULATE,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac,V1f2: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) )
!= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__div @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQ__CALCULATE,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac,V1f2: tp__ty_2Efrac_2Efrac] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) )
<=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Efrac_2Efrac__dnm @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) @ ( ap @ c_2Efrac_2Efrac__dnm @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__CALCULATE,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac,V1f2: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Efrac_2Efrac__dnm @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) @ ( ap @ c_2Efrac_2Efrac__dnm @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LEQ__CALCULATE,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac,V1f2: tp__ty_2Efrac_2Efrac] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) @ ( ap @ c_2Efrac_2Efrac__dnm @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Efrac_2Efrac__nmr @ ( inj__ty_2Efrac_2Efrac @ V1f2 ) ) ) @ ( ap @ c_2Efrac_2Efrac__dnm @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__OF__NUM__CALCULATE,axiom,
! [V0n1: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0n1 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Efrac_2Eabs__frac @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n1 ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__OF__NUM__LEQ,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__OF__NUM__LES,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQ0__NMR,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( V0r1
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__nmr @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__0LES__NMR,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Erat_2Erat__nmr @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES0__NMR,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ c_2Erat_2Erat__nmr @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__0LEQ__NMR,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Erat_2Erat__nmr @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LEQ0__NMR,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ c_2Erat_2Erat__nmr @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__ADD__ASSOC,axiom,
! [V0a: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat,V2c: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V1b ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__ASSOC,axiom,
! [V0a: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat,V2c: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V1b ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__ADD__COMM,axiom,
! [V0a: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V1b ) ) @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__COMM,axiom,
! [V0a: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V1b ) ) @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__ADD__RID,axiom,
! [V0a: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= V0a ) ).
thf(conj_thm_2Erat_2ERAT__ADD__LID,axiom,
! [V0a: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0a ) ) )
= V0a ) ).
thf(conj_thm_2Erat_2ERAT__MUL__RID,axiom,
! [V0a: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= V0a ) ).
thf(conj_thm_2Erat_2ERAT__MUL__LID,axiom,
! [V0a: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erat_2Erat @ V0a ) ) )
= V0a ) ).
thf(conj_thm_2Erat_2ERAT__ADD__RINV,axiom,
! [V0a: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__ADD__LINV,axiom,
! [V0a: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) @ ( inj__ty_2Erat_2Erat @ V0a ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__RINV,axiom,
! [V0a: tp__ty_2Erat_2Erat] :
( ( V0a
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__LINV,axiom,
! [V0a: tp__ty_2Erat_2Erat] :
( ( V0a
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) @ ( inj__ty_2Erat_2Erat @ V0a ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__RDISTRIB,axiom,
! [V0a: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat,V2c: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V1b ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LDISTRIB,axiom,
! [V0a: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat,V2c: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2c ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2c ) ) @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2c ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__1__NOT__0,axiom,
( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__LZERO,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__RZERO,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SUB__ADDAINV,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIV__MULMINV,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIV__0,axiom,
! [V0x: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0x ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__AINV__0,axiom,
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__AINV__AINV,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) )
= V0r1 ) ).
thf(conj_thm_2Erat_2ERAT__AINV__ADD,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__AINV__SUB,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__AINV__RMUL,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__AINV__LMUL,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__AINV__EQ,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= V1r2 )
<=> ( V0r1
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQ__AINV,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
<=> ( V0r1 = V1r2 ) ) ).
thf(conj_thm_2Erat_2ERAT__AINV__MINV,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( V0r1
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__minv @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SUB__RDISTRIB,axiom,
! [V0a: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat,V2c: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V1b ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SUB__LDISTRIB,axiom,
! [V0a: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat,V2c: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2c ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2c ) ) @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2c ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SUB__LID,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SUB__RID,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= V0r1 ) ).
thf(conj_thm_2Erat_2ERAT__SUB__ID,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0r ) ) @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQ__SUB0,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V0r1 = V1r2 ) ) ).
thf(conj_thm_2Erat_2ERAT__EQ__0SUB,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
<=> ( V0r1 = V1r2 ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__CALCULATE,axiom,
! [V0f1: tp__ty_2Efrac_2Efrac] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( ap @ c_2Erat_2Eabs__rat @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Efrac_2Efrac__sgn @ ( inj__ty_2Efrac_2Efrac @ V0f1 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__CLAUSES,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V0r1
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__gre @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__0,axiom,
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__AINV,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Erat_2Erat__sgn @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__MUL,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__MINV,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( V0r1
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__TOTAL,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
| ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__COMPLEMENT,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
!= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
<=> ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
!= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
| ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
!= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
<=> ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
| ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__REF,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
~ ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__ANTISYM,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
=> ~ ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__TRANS,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__TOTAL,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
| ( V0r1 = V1r2 )
| ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LEQ__REF,axiom,
! [V0r1: tp__ty_2Erat_2Erat] : ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LEQ__ANTISYM,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) )
=> ( V0r1 = V1r2 ) ) ).
thf(conj_thm_2Erat_2ERAT__LEQ__TRANS,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__01,axiom,
p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__IMP__LEQ,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__IMP__NEQ,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
=> ( V0r1 != V1r2 ) ) ).
thf(conj_thm_2Erat_2ERAT__LEQ__LES,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ~ ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__LEQ,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ~ ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__LEQ2,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
& ~ ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__LEQ__TRANS,axiom,
! [V0a: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat,V2c: tp__ty_2Erat_2Erat] :
( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) )
& ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V1b ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LEQ__LES__TRANS,axiom,
! [V0a: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat,V2c: tp__ty_2Erat_2Erat] :
( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) )
& ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1b ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__0LES__0LES__ADD,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES0__LES0__ADD,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__0LES__0LEQ__ADD,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES0__LEQ0__ADD,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LSUB__EQ,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= V2r3 )
<=> ( V0r1
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__RSUB__EQ,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( V0r1
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
= V1r2 ) ) ).
thf(conj_thm_2Erat_2ERAT__LDIV__EQ,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( V1r2
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= V2r3 )
<=> ( V0r1
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__RDIV__EQ,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( V2r3
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( V0r1
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
= V1r2 ) ) ) ).
thf(conj_thm_2Erat_2ERAT__AINV__ONE__ONE,axiom,
p @ ( ap @ ( c_2Ebool_2EONE__ONE @ ty_2Erat_2Erat @ ty_2Erat_2Erat ) @ c_2Erat_2Erat__ainv ) ).
thf(conj_thm_2Erat_2ERAT__ADD__ONE__ONE,axiom,
! [V0r1: tp__ty_2Erat_2Erat] : ( p @ ( ap @ ( c_2Ebool_2EONE__ONE @ ty_2Erat_2Erat @ ty_2Erat_2Erat ) @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__ONE__ONE,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( V0r1
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( c_2Ebool_2EONE__ONE @ ty_2Erat_2Erat @ ty_2Erat_2Erat ) @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQ__LADD,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
<=> ( V0r1 = V1r2 ) ) ).
thf(conj_thm_2Erat_2ERAT__EQ__RADD,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( V0r1 = V1r2 ) ) ).
thf(conj_thm_2Erat_2ERAT__EQ__RMUL,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( V2r3
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( V0r1 = V1r2 ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQ__LMUL,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( V2r3
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
<=> ( V0r1 = V1r2 ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__RADD,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__LADD,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LEQ__RADD,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LEQ__LADD,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__ADD__MONO,axiom,
! [V0a: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat,V2c: tp__ty_2Erat_2Erat,V3d: tp__ty_2Erat_2Erat] :
( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) )
& ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V2c ) ) @ ( inj__ty_2Erat_2Erat @ V3d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V2c ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V1b ) ) @ ( inj__ty_2Erat_2Erat @ V3d ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__AINV,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LSUB__LES,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__RSUB__LES,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LSUB__LEQ,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__RSUB__LEQ,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__RMUL__POS,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__LMUL__POS,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__RMUL__NEG,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__LMUL__NEG,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__AINV__LES,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LDIV__LES__POS,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LDIV__LES__NEG,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__RDIV__LES__POS,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__RDIV__LES__NEG,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LDIV__LEQ__POS,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LDIV__LEQ__NEG,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__RDIV__LEQ__POS,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__RDIV__LEQ__NEG,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__SUB0,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LES__0SUB,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__sub @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MINV__LES,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__SIGN__CASES,axiom,
! [V0p: tp__ty_2Erat_2Erat,V1q: tp__ty_2Erat_2Erat] :
( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0p ) ) @ ( inj__ty_2Erat_2Erat @ V1q ) ) ) )
<=> ( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0p ) ) )
& ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1q ) ) ) )
| ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0p ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1q ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0p ) ) @ ( inj__ty_2Erat_2Erat @ V1q ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0p ) ) )
& ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V1q ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
| ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0p ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1q ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__NO__ZERODIV,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( ( V0r1
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( V1r2
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
<=> ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__NO__ZERODIV__THM,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V0r1
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( V1r2
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__NO__ZERODIV__NEG,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V0r1
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1r2
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__NO__IDDIV,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= V1r2 )
<=> ( ( V0r1
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
| ( V1r2
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERDIV__MUL__OUT,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) ) ).
thf(conj_thm_2Erat_2ELDIV__MUL__OUT,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat,V2r3: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r3 ) ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DENSE__THM,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r3: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r3 ) ) )
=> ? [V2r2: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V2r2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V2r2 ) ) @ ( inj__ty_2Erat_2Erat @ V1r3 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SAVE,axiom,
! [V0r1: tp__ty_2Erat_2Erat] :
? [V1a1: tp__ty_2Einteger_2Eint,V2b1: tp__ty_2Enum_2Enum] :
( V0r1
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__save @ ( inj__ty_2Einteger_2Eint @ V1a1 ) ) @ ( inj__ty_2Enum_2Enum @ V2b1 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SAVE__MINV,axiom,
! [V0a1: tp__ty_2Einteger_2Eint,V1b1: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__save @ ( inj__ty_2Einteger_2Eint @ V0a1 ) ) @ ( inj__ty_2Enum_2Enum @ V1b1 ) ) ) )
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__minv @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__save @ ( inj__ty_2Einteger_2Eint @ V0a1 ) ) @ ( inj__ty_2Enum_2Enum @ V1b1 ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__save @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2EintExtension_2ESGN @ ( inj__ty_2Einteger_2Eint @ V0a1 ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V1b1 ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ c_2Einteger_2ENum @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ c_2Einteger_2EABS @ ( inj__ty_2Einteger_2Eint @ V0a1 ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SAVE__TO__CONS,axiom,
! [V0a1: tp__ty_2Einteger_2Eint,V1b1: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__save @ ( inj__ty_2Einteger_2Eint @ V0a1 ) ) @ ( inj__ty_2Enum_2Enum @ V1b1 ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__cons @ ( inj__ty_2Einteger_2Eint @ V0a1 ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V1b1 ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__OF__NUM,axiom,
! [A_27a: del,V0n: $i] :
( ( mem @ V0n @ A_27a )
=> ( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= fo__c_2Erat_2Erat__0 )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Erat_2Erat @ fo__c_2Erat_2Erat__1 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SAVE__NUM,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ ( ap @ c_2Efrac_2Efrac__save @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__CONS__TO__NUM,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__cons @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__cons @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__0,axiom,
( fo__c_2Erat_2Erat__0
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__1,axiom,
( fo__c_2Erat_2Erat__1
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MINV__1,axiom,
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__minv @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIV__1,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0r ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= V0r ) ).
thf(conj_thm_2Erat_2ERAT__DIV__NEG1,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0r ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIV__INV,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( V0r
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0r ) ) @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MINV__MUL,axiom,
! [V0a: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat] :
( ( ( V0a
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1b
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__minv @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ V0a ) ) ) @ ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIVDIV__MUL,axiom,
! [V0b: tp__ty_2Erat_2Erat,V1d: tp__ty_2Erat_2Erat,V2a: tp__ty_2Erat_2Erat,V3c: tp__ty_2Erat_2Erat] :
( ( ( V0b
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1d
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V2a ) ) @ ( inj__ty_2Erat_2Erat @ V0b ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V3c ) ) @ ( inj__ty_2Erat_2Erat @ V1d ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2a ) ) @ ( inj__ty_2Erat_2Erat @ V3c ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0b ) ) @ ( inj__ty_2Erat_2Erat @ V1d ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIVDIV__ADD,axiom,
! [V0y: tp__ty_2Erat_2Erat,V1b: tp__ty_2Erat_2Erat,V2x: tp__ty_2Erat_2Erat,V3a: tp__ty_2Erat_2Erat] :
( ( ( V0y
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1b
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V2x ) ) @ ( inj__ty_2Erat_2Erat @ V0y ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V3a ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V2x ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V3a ) ) @ ( inj__ty_2Erat_2Erat @ V0y ) ) ) ) @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0y ) ) @ ( inj__ty_2Erat_2Erat @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIV__AINV,axiom,
! [V0x: tp__ty_2Erat_2Erat,V1y: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0x ) ) @ ( inj__ty_2Erat_2Erat @ V1y ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0x ) ) ) @ ( inj__ty_2Erat_2Erat @ V1y ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MINV__EQ__0,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( V0r
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIV__MINV,axiom,
! [V0x: tp__ty_2Erat_2Erat,V1y: tp__ty_2Erat_2Erat] :
( ( ( V0x
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1y
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__minv @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V0x ) ) @ ( inj__ty_2Erat_2Erat @ V1y ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V1y ) ) @ ( inj__ty_2Erat_2Erat @ V0x ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__DIV__EQ0,axiom,
! [V0d: tp__ty_2Erat_2Erat,V1n: tp__ty_2Erat_2Erat] :
( ( V0d
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V1n ) ) @ ( inj__ty_2Erat_2Erat @ V0d ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V1n
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V1n ) ) @ ( inj__ty_2Erat_2Erat @ V0d ) ) ) )
<=> ( V1n
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__ADD__NUM__CALCULATE,axiom,
( ! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erat_2Erat ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum,V5m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erat_2Erat ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) )
& ! [V6n: tp__ty_2Enum_2Enum,V7m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V7m ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( inj__ty_2Enum_2Enum @ V7m ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MUL__NUM__CALCULATE,axiom,
( ! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum,V5m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) ) ) )
& ! [V6n: tp__ty_2Enum_2Enum,V7m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V7m ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( inj__ty_2Enum_2Enum @ V7m ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__EQ__NUM__CALCULATE,axiom,
( ! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ( V0n = V1m ) )
& ! [V2n: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
<=> ( ( V2n = fo__c_2Enum_2E0 )
& ( V3m = fo__c_2Enum_2E0 ) ) )
& ! [V4n: tp__ty_2Enum_2Enum,V5m: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) )
<=> ( ( V4n = fo__c_2Enum_2E0 )
& ( V5m = fo__c_2Enum_2E0 ) ) )
& ! [V6n: tp__ty_2Enum_2Enum,V7m: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V7m ) ) ) ) )
<=> ( V6n = V7m ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LT__NUM__CALCULATE,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
| ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__LE__NUM__CALCULATE,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
<=> ( ( V2m = fo__c_2Enum_2E0 )
& ( V3n = fo__c_2Enum_2E0 ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__of__int__def,axiom,
! [V0i: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V0i ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erat_2Erat ) @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( inj__ty_2Einteger_2Eint @ V0i ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Einteger_2ENum @ ( ap @ c_2Einteger_2Eint__neg @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Einteger_2ENum @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__of__int__11,axiom,
! [V0i1: tp__ty_2Einteger_2Eint,V1i2: tp__ty_2Einteger_2Eint] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V0i1 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V1i2 ) ) ) )
<=> ( V0i1 = V1i2 ) ) ).
thf(conj_thm_2Erat_2Erat__of__int__of__num,axiom,
! [V0x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__int @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__of__int__MUL,axiom,
! [V0x: tp__ty_2Einteger_2Eint,V1y: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V0x ) ) ) @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V1y ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__int @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__ty_2Einteger_2Eint @ V1y ) ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__of__int__ADD,axiom,
! [V0x: tp__ty_2Einteger_2Eint,V1y: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__add @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V0x ) ) ) @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V1y ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__int @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__ty_2Einteger_2Eint @ V1y ) ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__of__int__LE,axiom,
! [V0i: tp__ty_2Einteger_2Eint,V1j: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V1j ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( inj__ty_2Einteger_2Eint @ V0i ) ) @ ( inj__ty_2Einteger_2Eint @ V1j ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__of__int__LT,axiom,
! [V0i: tp__ty_2Einteger_2Eint,V1j: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V1j ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( inj__ty_2Einteger_2Eint @ V0i ) ) @ ( inj__ty_2Einteger_2Eint @ V1j ) ) ) ) ).
thf(conj_thm_2Erat_2Erat__of__int__ainv,axiom,
! [V0i: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__int @ ( ap @ c_2Einteger_2Eint__neg @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__OF__INT__CALCULATE,axiom,
! [V0i: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V0i ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Eabs__rat @ ( ap @ c_2Efrac_2Eabs__frac @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) @ ( inj__ty_2Einteger_2Eint @ V0i ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erat_2ERATND__THM,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( V0r
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ c_2Erat_2Erat__of__int @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Erat_2ERATD @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Erat_2ERATD @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) )
& ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Erat_2ERATD @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V1n_27: tp__ty_2Einteger_2Eint,V2d_27: tp__ty_2Enum_2Enum] :
( ( ( V0r
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V1n_27 ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V2d_27 ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2d_27 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ c_2Einteger_2EABS @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) @ ( ap @ c_2Einteger_2EABS @ ( inj__ty_2Einteger_2Eint @ V1n_27 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERATD__NZERO,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Erat_2ERATD @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Erat_2ERATD @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
!= fo__c_2Enum_2E0 ) ) ).
thf(conj_thm_2Erat_2ERATN__LEAST,axiom,
! [V0r: tp__ty_2Erat_2Erat,V1n_27: tp__ty_2Einteger_2Eint,V2d_27: tp__ty_2Enum_2Enum] :
( ( ( V0r
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V1n_27 ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V2d_27 ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2d_27 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ c_2Einteger_2EABS @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) @ ( ap @ c_2Einteger_2EABS @ ( inj__ty_2Einteger_2Eint @ V1n_27 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERATN__RATD__EQ__THM,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( V0r
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ c_2Erat_2Erat__of__int @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Erat_2ERATD @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERATN__RATD__MULT,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( inj__ty_2Erat_2Erat @ V0r ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Erat_2ERATD @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__int @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERATND__RAT__OF__NUM,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2ERATN @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Erat_2ERATD @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERATN__EQ0,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V0r
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) )
<=> ( V0r
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERATN__SIGN,axiom,
! [V0x: tp__ty_2Erat_2Erat] :
( ( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0x ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0x ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0x ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0x ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0x ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0x ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0x ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__leq @ ( inj__ty_2Erat_2Erat @ V0x ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__AINV__SGN,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERATN__NEG,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2ERATN @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERATD__NEG,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Erat_2ERATD @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Erat_2ERATD @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ).
thf(conj_thm_2Erat_2ERATN__RATD__RAT__OF__INT,axiom,
! [V0i: tp__ty_2Einteger_2Eint] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2ERATN @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) )
= V0i )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Erat_2ERATD @ ( ap @ c_2Erat_2Erat__of__int @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERATN__DIV__RATD,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ c_2Erat_2Erat__of__int @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Erat_2ERATD @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) )
= V0r ) ).
thf(conj_thm_2Erat_2ERAT__AINV__EQ__NUM,axiom,
! [V0x: tp__ty_2Erat_2Erat,V1n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0x ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( V0x
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__int @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__NUM__COND,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Einteger_2Eint ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__AINV__RWT,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( ap @ c_2Erat_2Erat__ainv @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__ALT,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2EintExtension_2ESGN @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__NUM__BITs,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__EQ0,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V0r
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) )
<=> ( V0r
= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__POS,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__NEG,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__SGN__DIV,axiom,
! [V0d: tp__ty_2Erat_2Erat,V1n: tp__ty_2Erat_2Erat] :
( ( V0d
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Erat_2Erat__sgn @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( inj__ty_2Erat_2Erat @ V1n ) ) @ ( inj__ty_2Erat_2Erat @ V0d ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V1n ) ) ) @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0d ) ) ) ) ) ) ).
thf(conj_thm_2Erat_2ERAT__MINV__RATND,axiom,
! [V0r: tp__ty_2Erat_2Erat] :
( ( V0r
!= ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erat_2Erat @ ( ap @ c_2Erat_2Erat__minv @ ( inj__ty_2Erat_2Erat @ V0r ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__div @ ( ap @ ( ap @ c_2Erat_2Erat__mul @ ( ap @ c_2Erat_2Erat__of__int @ ( ap @ c_2Erat_2Erat__sgn @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) @ ( ap @ c_2Erat_2Erat__of__num @ ( ap @ c_2Erat_2ERATD @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ) @ ( ap @ c_2Erat_2Erat__of__int @ ( ap @ c_2Einteger_2EABS @ ( ap @ c_2Erat_2ERATN @ ( inj__ty_2Erat_2Erat @ V0r ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__min__def,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__min @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erat_2Erat ) @ ( ap @ ( ap @ c_2Erat_2Erat__les @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
thf(ax_thm_2Erat_2Erat__max__def,axiom,
! [V0r1: tp__ty_2Erat_2Erat,V1r2: tp__ty_2Erat_2Erat] :
( ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ c_2Erat_2Erat__max @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) )
= ( surj__ty_2Erat_2Erat @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erat_2Erat ) @ ( ap @ ( ap @ c_2Erat_2Erat__gre @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) @ ( inj__ty_2Erat_2Erat @ V0r1 ) ) @ ( inj__ty_2Erat_2Erat @ V1r2 ) ) ) ) ).
%------------------------------------------------------------------------------