ITP001 Axioms: ITP099^5.ax
%------------------------------------------------------------------------------
% File : ITP099^5 : 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 : hreal^2.ax [Gau20]
% : HL4099^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 113 ( 26 unt; 21 typ; 0 def)
% Number of atoms : 1059 ( 44 equ; 0 cnn)
% Maximal formula atoms : 34 ( 9 avg)
% Number of connectives : 1664 ( 14 ~; 6 |; 24 &;1574 @)
% ( 18 <=>; 28 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg;1574 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 9 ( 9 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 42 usr; 36 con; 0-2 aty)
% Number of variables : 207 ( 27 ^ 157 !; 23 ?; 207 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Ehreal_2Ehreal,type,
ty_2Ehreal_2Ehreal: del ).
thf(stp_ty_2Ehreal_2Ehreal,type,
tp__ty_2Ehreal_2Ehreal: $tType ).
thf(stp_inj_ty_2Ehreal_2Ehreal,type,
inj__ty_2Ehreal_2Ehreal: tp__ty_2Ehreal_2Ehreal > $i ).
thf(stp_surj_ty_2Ehreal_2Ehreal,type,
surj__ty_2Ehreal_2Ehreal: $i > tp__ty_2Ehreal_2Ehreal ).
thf(stp_inj_surj_ty_2Ehreal_2Ehreal,axiom,
! [X: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( inj__ty_2Ehreal_2Ehreal @ X ) )
= X ) ).
thf(stp_inj_mem_ty_2Ehreal_2Ehreal,axiom,
! [X: tp__ty_2Ehreal_2Ehreal] : ( mem @ ( inj__ty_2Ehreal_2Ehreal @ X ) @ ty_2Ehreal_2Ehreal ) ).
thf(stp_iso_mem_ty_2Ehreal_2Ehreal,axiom,
! [X: $i] :
( ( mem @ X @ ty_2Ehreal_2Ehreal )
=> ( X
= ( inj__ty_2Ehreal_2Ehreal @ ( surj__ty_2Ehreal_2Ehreal @ X ) ) ) ) ).
thf(tp_c_2Ehreal_2Ecut,type,
c_2Ehreal_2Ecut: $i ).
thf(mem_c_2Ehreal_2Ecut,axiom,
mem @ c_2Ehreal_2Ecut @ ( arr @ ty_2Ehreal_2Ehreal @ ( arr @ ty_2Ehrat_2Ehrat @ bool ) ) ).
thf(tp_c_2Ehreal_2Ecut__of__hrat,type,
c_2Ehreal_2Ecut__of__hrat: $i ).
thf(mem_c_2Ehreal_2Ecut__of__hrat,axiom,
mem @ c_2Ehreal_2Ecut__of__hrat @ ( arr @ ty_2Ehrat_2Ehrat @ ( arr @ ty_2Ehrat_2Ehrat @ bool ) ) ).
thf(tp_c_2Ehreal_2Ehrat__lt,type,
c_2Ehreal_2Ehrat__lt: $i ).
thf(mem_c_2Ehreal_2Ehrat__lt,axiom,
mem @ c_2Ehreal_2Ehrat__lt @ ( arr @ ty_2Ehrat_2Ehrat @ ( arr @ ty_2Ehrat_2Ehrat @ bool ) ) ).
thf(tp_c_2Ehreal_2Ehreal,type,
c_2Ehreal_2Ehreal: $i ).
thf(mem_c_2Ehreal_2Ehreal,axiom,
mem @ c_2Ehreal_2Ehreal @ ( arr @ ( arr @ ty_2Ehrat_2Ehrat @ bool ) @ ty_2Ehreal_2Ehreal ) ).
thf(tp_c_2Ehreal_2Ehreal__1,type,
c_2Ehreal_2Ehreal__1: $i ).
thf(mem_c_2Ehreal_2Ehreal__1,axiom,
mem @ c_2Ehreal_2Ehreal__1 @ ty_2Ehreal_2Ehreal ).
thf(stp_fo_c_2Ehreal_2Ehreal__1,type,
fo__c_2Ehreal_2Ehreal__1: tp__ty_2Ehreal_2Ehreal ).
thf(stp_eq_fo_c_2Ehreal_2Ehreal__1,axiom,
( ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 )
= c_2Ehreal_2Ehreal__1 ) ).
thf(tp_c_2Ehreal_2Ehreal__add,type,
c_2Ehreal_2Ehreal__add: $i ).
thf(mem_c_2Ehreal_2Ehreal__add,axiom,
mem @ c_2Ehreal_2Ehreal__add @ ( arr @ ty_2Ehreal_2Ehreal @ ( arr @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) ).
thf(stp_fo_c_2Ehreal_2Ehreal__add,type,
fo__c_2Ehreal_2Ehreal__add: tp__ty_2Ehreal_2Ehreal > tp__ty_2Ehreal_2Ehreal > tp__ty_2Ehreal_2Ehreal ).
thf(stp_eq_fo_c_2Ehreal_2Ehreal__add,axiom,
! [X0: tp__ty_2Ehreal_2Ehreal,X1: tp__ty_2Ehreal_2Ehreal] :
( ( inj__ty_2Ehreal_2Ehreal @ ( fo__c_2Ehreal_2Ehreal__add @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ X0 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ X1 ) ) ) ).
thf(tp_c_2Ehreal_2Ehreal__inv,type,
c_2Ehreal_2Ehreal__inv: $i ).
thf(mem_c_2Ehreal_2Ehreal__inv,axiom,
mem @ c_2Ehreal_2Ehreal__inv @ ( arr @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ).
thf(stp_fo_c_2Ehreal_2Ehreal__inv,type,
fo__c_2Ehreal_2Ehreal__inv: tp__ty_2Ehreal_2Ehreal > tp__ty_2Ehreal_2Ehreal ).
thf(stp_eq_fo_c_2Ehreal_2Ehreal__inv,axiom,
! [X0: tp__ty_2Ehreal_2Ehreal] :
( ( inj__ty_2Ehreal_2Ehreal @ ( fo__c_2Ehreal_2Ehreal__inv @ X0 ) )
= ( ap @ c_2Ehreal_2Ehreal__inv @ ( inj__ty_2Ehreal_2Ehreal @ X0 ) ) ) ).
thf(tp_c_2Ehreal_2Ehreal__lt,type,
c_2Ehreal_2Ehreal__lt: $i ).
thf(mem_c_2Ehreal_2Ehreal__lt,axiom,
mem @ c_2Ehreal_2Ehreal__lt @ ( arr @ ty_2Ehreal_2Ehreal @ ( arr @ ty_2Ehreal_2Ehreal @ bool ) ) ).
thf(tp_c_2Ehreal_2Ehreal__mul,type,
c_2Ehreal_2Ehreal__mul: $i ).
thf(mem_c_2Ehreal_2Ehreal__mul,axiom,
mem @ c_2Ehreal_2Ehreal__mul @ ( arr @ ty_2Ehreal_2Ehreal @ ( arr @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) ).
thf(stp_fo_c_2Ehreal_2Ehreal__mul,type,
fo__c_2Ehreal_2Ehreal__mul: tp__ty_2Ehreal_2Ehreal > tp__ty_2Ehreal_2Ehreal > tp__ty_2Ehreal_2Ehreal ).
thf(stp_eq_fo_c_2Ehreal_2Ehreal__mul,axiom,
! [X0: tp__ty_2Ehreal_2Ehreal,X1: tp__ty_2Ehreal_2Ehreal] :
( ( inj__ty_2Ehreal_2Ehreal @ ( fo__c_2Ehreal_2Ehreal__mul @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ X0 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ X1 ) ) ) ).
thf(tp_c_2Ehreal_2Ehreal__sub,type,
c_2Ehreal_2Ehreal__sub: $i ).
thf(mem_c_2Ehreal_2Ehreal__sub,axiom,
mem @ c_2Ehreal_2Ehreal__sub @ ( arr @ ty_2Ehreal_2Ehreal @ ( arr @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) ).
thf(stp_fo_c_2Ehreal_2Ehreal__sub,type,
fo__c_2Ehreal_2Ehreal__sub: tp__ty_2Ehreal_2Ehreal > tp__ty_2Ehreal_2Ehreal > tp__ty_2Ehreal_2Ehreal ).
thf(stp_eq_fo_c_2Ehreal_2Ehreal__sub,axiom,
! [X0: tp__ty_2Ehreal_2Ehreal,X1: tp__ty_2Ehreal_2Ehreal] :
( ( inj__ty_2Ehreal_2Ehreal @ ( fo__c_2Ehreal_2Ehreal__sub @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Ehreal_2Ehreal__sub @ ( inj__ty_2Ehreal_2Ehreal @ X0 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ X1 ) ) ) ).
thf(tp_c_2Ehreal_2Ehreal__sup,type,
c_2Ehreal_2Ehreal__sup: $i ).
thf(mem_c_2Ehreal_2Ehreal__sup,axiom,
mem @ c_2Ehreal_2Ehreal__sup @ ( arr @ ( arr @ ty_2Ehreal_2Ehreal @ bool ) @ ty_2Ehreal_2Ehreal ) ).
thf(tp_c_2Ehreal_2Eisacut,type,
c_2Ehreal_2Eisacut: $i ).
thf(mem_c_2Ehreal_2Eisacut,axiom,
mem @ c_2Ehreal_2Eisacut @ ( arr @ ( arr @ ty_2Ehrat_2Ehrat @ bool ) @ bool ) ).
thf(ax_thm_2Ehreal_2Ehrat__lt,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) )
<=> ? [V2d: tp__ty_2Ehrat_2Ehrat] :
( V1y
= ( surj__ty_2Ehrat_2Ehrat @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2d ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__REFL,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat] :
~ ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__TRANS,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat,V2z: tp__ty_2Ehrat_2Ehrat] :
( ( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__ANTISYM,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
~ ( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__TOTAL,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
( ( V0x = V1y )
| ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) )
| ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__MUL__RID,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat] :
( ( surj__ty_2Ehrat_2Ehrat @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ fo__c_2Ehrat_2Ehrat__1 ) ) )
= V0x ) ).
thf(conj_thm_2Ehreal_2EHRAT__MUL__RINV,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat] :
( ( surj__ty_2Ehrat_2Ehrat @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( ap @ c_2Ehrat_2Ehrat__inv @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) )
= fo__c_2Ehrat_2Ehrat__1 ) ).
thf(conj_thm_2Ehreal_2EHRAT__RDISTRIB,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat,V2z: tp__ty_2Ehrat_2Ehrat] :
( ( surj__ty_2Ehrat_2Ehrat @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) )
= ( surj__ty_2Ehrat_2Ehrat @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__ADDL,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] : ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__ADDR,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] : ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__GT,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) )
=> ~ ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__NE,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) )
=> ( V0x != V1y ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__EQ__LADD,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat,V2z: tp__ty_2Ehrat_2Ehrat] :
( ( ( surj__ty_2Ehrat_2Ehrat @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) )
= ( surj__ty_2Ehrat_2Ehrat @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) ) )
<=> ( V1y = V2z ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__EQ__LMUL,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat,V2z: tp__ty_2Ehrat_2Ehrat] :
( ( ( surj__ty_2Ehrat_2Ehrat @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) )
= ( surj__ty_2Ehrat_2Ehrat @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) ) )
<=> ( V1y = V2z ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__ADD2,axiom,
! [V0u: tp__ty_2Ehrat_2Ehrat,V1v: tp__ty_2Ehrat_2Ehrat,V2x: tp__ty_2Ehrat_2Ehrat,V3y: tp__ty_2Ehrat_2Ehrat] :
( ( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0u ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1v ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V3y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V0u ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1v ) ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V3y ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__LADD,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat,V2z: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__RADD,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat,V2z: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__MUL2,axiom,
! [V0u: tp__ty_2Ehrat_2Ehrat,V1v: tp__ty_2Ehrat_2Ehrat,V2x: tp__ty_2Ehrat_2Ehrat,V3y: tp__ty_2Ehrat_2Ehrat] :
( ( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0u ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1v ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V3y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V0u ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1v ) ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V3y ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__LMUL,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat,V2z: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__RMUL,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat,V2z: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__LMUL1,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ fo__c_2Ehrat_2Ehrat__1 ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__RMUL1,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ fo__c_2Ehrat_2Ehrat__1 ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__GT__LMUL1,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ fo__c_2Ehrat_2Ehrat__1 ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__L1,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( ap @ c_2Ehrat_2Ehrat__inv @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) @ ( inj__ty_2Ehrat_2Ehrat @ fo__c_2Ehrat_2Ehrat__1 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__LT__R1,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( ap @ c_2Ehrat_2Ehrat__inv @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ) @ ( inj__ty_2Ehrat_2Ehrat @ fo__c_2Ehrat_2Ehrat__1 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__GT__L1,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ fo__c_2Ehrat_2Ehrat__1 ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( ap @ c_2Ehrat_2Ehrat__inv @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__INV__MUL,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
( ( surj__ty_2Ehrat_2Ehrat @ ( ap @ c_2Ehrat_2Ehrat__inv @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) )
= ( surj__ty_2Ehrat_2Ehrat @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( ap @ c_2Ehrat_2Ehrat__inv @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) @ ( ap @ c_2Ehrat_2Ehrat__inv @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__UP,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat] :
? [V1y: tp__ty_2Ehrat_2Ehrat] : ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__DOWN,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat] :
? [V1y: tp__ty_2Ehrat_2Ehrat] : ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__DOWN2,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
? [V2z: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHRAT__MEAN,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat,V1y: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) )
=> ? [V2z: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V2z ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1y ) ) ) ) ) ).
thf(ax_thm_2Ehreal_2Eisacut,axiom,
! [V0C: $i] :
( ( mem @ V0C @ ( arr @ ty_2Ehrat_2Ehrat @ bool ) )
=> ( ( p @ ( ap @ c_2Ehreal_2Eisacut @ V0C ) )
<=> ( ? [V1x: tp__ty_2Ehrat_2Ehrat] : ( p @ ( ap @ V0C @ ( inj__ty_2Ehrat_2Ehrat @ V1x ) ) )
& ? [V2x: tp__ty_2Ehrat_2Ehrat] :
~ ( p @ ( ap @ V0C @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) )
& ! [V3x: tp__ty_2Ehrat_2Ehrat,V4y: tp__ty_2Ehrat_2Ehrat] :
( ( ( p @ ( ap @ V0C @ ( inj__ty_2Ehrat_2Ehrat @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V4y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V3x ) ) ) )
=> ( p @ ( ap @ V0C @ ( inj__ty_2Ehrat_2Ehrat @ V4y ) ) ) )
& ! [V5x: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ V0C @ ( inj__ty_2Ehrat_2Ehrat @ V5x ) ) )
=> ? [V6y: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ V0C @ ( inj__ty_2Ehrat_2Ehrat @ V6y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V5x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V6y ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ehreal_2Ecut__of__hrat,axiom,
! [V0x: tp__ty_2Ehrat_2Ehrat] :
( ( ap @ c_2Ehreal_2Ecut__of__hrat @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) )
= ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V1y: $i] : ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ V1y ) @ ( inj__ty_2Ehrat_2Ehrat @ V0x ) ) ) ) ).
thf(conj_thm_2Ehreal_2EISACUT__HRAT,axiom,
! [V0h: tp__ty_2Ehrat_2Ehrat] : ( p @ ( ap @ c_2Ehreal_2Eisacut @ ( ap @ c_2Ehreal_2Ecut__of__hrat @ ( inj__ty_2Ehrat_2Ehrat @ V0h ) ) ) ) ).
thf(ax_thm_2Ehreal_2Ehreal__TY__DEF,axiom,
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ty_2Ehreal_2Ehreal @ ( arr @ ty_2Ehrat_2Ehrat @ bool ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ty_2Ehrat_2Ehrat @ bool ) @ ty_2Ehreal_2Ehreal ) @ c_2Ehreal_2Eisacut ) @ V0rep ) ) ) ).
thf(ax_thm_2Ehreal_2Ehreal__tybij,axiom,
( ! [V0a: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ c_2Ehreal_2Ehreal @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0a ) ) ) )
= V0a )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ty_2Ehrat_2Ehrat @ bool ) )
=> ( ( p @ ( ap @ c_2Ehreal_2Eisacut @ V1r ) )
<=> ( ( ap @ c_2Ehreal_2Ecut @ ( ap @ c_2Ehreal_2Ehreal @ V1r ) )
= V1r ) ) ) ) ).
thf(conj_thm_2Ehreal_2EEQUAL__CUTS,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) )
= ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) )
=> ( V0X = V1Y ) ) ).
thf(conj_thm_2Ehreal_2ECUT__ISACUT,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal] : ( p @ ( ap @ c_2Ehreal_2Eisacut @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) ) ) ).
thf(conj_thm_2Ehreal_2ECUT__NONEMPTY,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal] :
? [V1x: tp__ty_2Ehrat_2Ehrat] : ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1x ) ) ) ).
thf(conj_thm_2Ehreal_2ECUT__BOUNDED,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal] :
? [V1x: tp__ty_2Ehrat_2Ehrat] :
~ ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1x ) ) ) ).
thf(conj_thm_2Ehreal_2ECUT__DOWN,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1x: tp__ty_2Ehrat_2Ehrat,V2y: tp__ty_2Ehrat_2Ehrat] :
( ( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V2y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1x ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2y ) ) ) ) ).
thf(conj_thm_2Ehreal_2ECUT__UP,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1x: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1x ) ) )
=> ? [V2y: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2y ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2ECUT__UBOUND,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1x: tp__ty_2Ehrat_2Ehrat,V2y: tp__ty_2Ehrat_2Ehrat] :
( ( ~ ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2y ) ) ) )
=> ~ ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2y ) ) ) ) ).
thf(conj_thm_2Ehreal_2ECUT__STRADDLE,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1x: tp__ty_2Ehrat_2Ehrat,V2y: tp__ty_2Ehrat_2Ehrat] :
( ( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1x ) ) )
& ~ ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ V1x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2y ) ) ) ) ).
thf(conj_thm_2Ehreal_2ECUT__NEARTOP__ADD,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1e: tp__ty_2Ehrat_2Ehrat] :
? [V2x: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) )
& ~ ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1e ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2ECUT__NEARTOP__MUL,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1u: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( inj__ty_2Ehrat_2Ehrat @ fo__c_2Ehrat_2Ehrat__1 ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V1u ) ) )
=> ? [V2x: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) )
& ~ ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ ( inj__ty_2Ehrat_2Ehrat @ V1u ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) ) ) ) ) ).
thf(ax_thm_2Ehreal_2Ehreal__1,axiom,
( fo__c_2Ehreal_2Ehreal__1
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ c_2Ehreal_2Ehreal @ ( ap @ c_2Ehreal_2Ecut__of__hrat @ ( inj__ty_2Ehrat_2Ehrat @ fo__c_2Ehrat_2Ehrat__1 ) ) ) ) ) ).
thf(ax_thm_2Ehreal_2Ehreal__add,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) )
= ( surj__ty_2Ehreal_2Ehreal
@ ( ap @ c_2Ehreal_2Ehreal
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V2w: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V3x: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V4y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Ehrat_2Ehrat ) @ V2w ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ V3x ) @ V4y ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ V3x ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ V4y ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ehreal_2Ehreal__mul,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) )
= ( surj__ty_2Ehreal_2Ehreal
@ ( ap @ c_2Ehreal_2Ehreal
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V2w: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V3x: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V4y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Ehrat_2Ehrat ) @ V2w ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ V3x ) @ V4y ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ V3x ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ V4y ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ehreal_2Ehreal__inv,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ c_2Ehreal_2Ehreal__inv @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) )
= ( surj__ty_2Ehreal_2Ehreal
@ ( ap @ c_2Ehreal_2Ehreal
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V1w: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V2d: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ V2d ) @ ( inj__ty_2Ehrat_2Ehrat @ fo__c_2Ehrat_2Ehrat__1 ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ V3x ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ V1w ) @ V3x ) ) @ V2d ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ehreal_2Ehreal__sup,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Ehreal_2Ehreal @ bool ) )
=> ( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ c_2Ehreal_2Ehreal__sup @ V0P ) )
= ( surj__ty_2Ehreal_2Ehreal
@ ( ap @ c_2Ehreal_2Ehreal
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V1w: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehreal_2Ehreal )
@ ( lam @ ty_2Ehreal_2Ehreal
@ ^ [V2X: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V2X ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ V2X ) @ V1w ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ehreal_2Ehreal__lt,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) )
<=> ( ( V0X != V1Y )
& ! [V2x: tp__ty_2Ehrat_2Ehrat] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__INV__ISACUT,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal] :
( p
@ ( ap @ c_2Ehreal_2Eisacut
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V1w: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V2d: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ V2d ) @ ( inj__ty_2Ehrat_2Ehrat @ fo__c_2Ehrat_2Ehrat__1 ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ V3x ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehrat__lt @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ V1w ) @ V3x ) ) @ V2d ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__ADD__ISACUT,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( p
@ ( ap @ c_2Ehreal_2Eisacut
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V2w: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V3x: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V4y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Ehrat_2Ehrat ) @ V2w ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ V3x ) @ V4y ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ V3x ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__MUL__ISACUT,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( p
@ ( ap @ c_2Ehreal_2Eisacut
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V2w: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V3x: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V4y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Ehrat_2Ehrat ) @ V2w ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__mul @ V3x ) @ V4y ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ V3x ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__ADD__SYM,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) )
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__MUL__SYM,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) )
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__ADD__ASSOC,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal,V2Z: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2Z ) ) ) )
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2Z ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__MUL__ASSOC,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal,V2Z: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2Z ) ) ) )
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2Z ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__LDISTRIB,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal,V2Z: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2Z ) ) ) )
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2Z ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__MUL__LID,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) )
= V0X ) ).
thf(conj_thm_2Ehreal_2EHREAL__MUL__LINV,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( ap @ c_2Ehreal_2Ehreal__inv @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) )
= fo__c_2Ehreal_2Ehreal__1 ) ).
thf(conj_thm_2Ehreal_2EHREAL__NOZERO,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) )
!= V0X ) ).
thf(ax_thm_2Ehreal_2Ehreal__sub,axiom,
! [V0Y: tp__ty_2Ehreal_2Ehreal,V1X: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__sub @ ( inj__ty_2Ehreal_2Ehreal @ V0Y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1X ) ) )
= ( surj__ty_2Ehreal_2Ehreal
@ ( ap @ c_2Ehreal_2Ehreal
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V2w: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V1X ) ) @ V3x ) ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0Y ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ V3x ) @ V2w ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__LT__LEMMA,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) )
=> ? [V2x: tp__ty_2Ehrat_2Ehrat] :
( ~ ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ ( inj__ty_2Ehrat_2Ehrat @ V2x ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__SUB__ISACUT,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) )
=> ( p
@ ( ap @ c_2Ehreal_2Eisacut
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V2w: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehrat_2Ehrat )
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ V3x ) ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ ( ap @ ( ap @ c_2Ehrat_2Ehrat__add @ V3x ) @ V2w ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__SUB__ADD,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) )
=> ( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__sub @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) )
= V1Y ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__LT__TOTAL,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( V0X = V1Y )
| ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) )
| ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__LT,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) )
<=> ? [V2D: tp__ty_2Ehreal_2Ehreal] :
( V1Y
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2D ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__ADD__TOTAL,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal,V1Y: tp__ty_2Ehreal_2Ehreal] :
( ( V0X = V1Y )
| ? [V2D: tp__ty_2Ehreal_2Ehreal] :
( V1Y
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2D ) ) ) )
| ? [V3D: tp__ty_2Ehreal_2Ehreal] :
( V0X
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V1Y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V3D ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__SUP__ISACUT,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Ehreal_2Ehreal @ bool ) )
=> ( ( ? [V1X: tp__ty_2Ehreal_2Ehreal] : ( p @ ( ap @ V0P @ ( inj__ty_2Ehreal_2Ehreal @ V1X ) ) )
& ? [V2Y: tp__ty_2Ehreal_2Ehreal] :
! [V3X: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Ehreal_2Ehreal @ V3X ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V3X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2Y ) ) ) ) )
=> ( p
@ ( ap @ c_2Ehreal_2Eisacut
@ ( lam @ ty_2Ehrat_2Ehrat
@ ^ [V4w: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Ehreal_2Ehreal )
@ ( lam @ ty_2Ehreal_2Ehreal
@ ^ [V5X: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V5X ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ecut @ V5X ) @ V4w ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ehreal_2EHREAL__SUP,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Ehreal_2Ehreal @ bool ) )
=> ( ( ? [V1X: tp__ty_2Ehreal_2Ehreal] : ( p @ ( ap @ V0P @ ( inj__ty_2Ehreal_2Ehreal @ V1X ) ) )
& ? [V2Y: tp__ty_2Ehreal_2Ehreal] :
! [V3X: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Ehreal_2Ehreal @ V3X ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V3X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2Y ) ) ) ) )
=> ! [V4Y: tp__ty_2Ehreal_2Ehreal] :
( ? [V5X: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Ehreal_2Ehreal @ V5X ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V4Y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V5X ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V4Y ) ) @ ( ap @ c_2Ehreal_2Ehreal__sup @ V0P ) ) ) ) ) ) ).
%------------------------------------------------------------------------------