ITP001 Axioms: ITP113^5.ax
%------------------------------------------------------------------------------
% File : ITP113^5 : 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 : realax^2.ax [Gau20]
% : HL4113^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 171 ( 51 unt; 40 typ; 0 def)
% Number of atoms : 1261 ( 66 equ; 0 cnn)
% Maximal formula atoms : 45 ( 7 avg)
% Number of connectives : 2144 ( 13 ~; 4 |; 22 &;2046 @)
% ( 17 <=>; 42 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg;2046 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 14 ( 14 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 64 usr; 52 con; 0-2 aty)
% Number of variables : 219 ( 8 ^ 199 !; 12 ?; 219 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Erealax_2Ereal,type,
ty_2Erealax_2Ereal: del ).
thf(stp_ty_2Erealax_2Ereal,type,
tp__ty_2Erealax_2Ereal: $tType ).
thf(stp_inj_ty_2Erealax_2Ereal,type,
inj__ty_2Erealax_2Ereal: tp__ty_2Erealax_2Ereal > $i ).
thf(stp_surj_ty_2Erealax_2Ereal,type,
surj__ty_2Erealax_2Ereal: $i > tp__ty_2Erealax_2Ereal ).
thf(stp_inj_surj_ty_2Erealax_2Ereal,axiom,
! [X: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( inj__ty_2Erealax_2Ereal @ X ) )
= X ) ).
thf(stp_inj_mem_ty_2Erealax_2Ereal,axiom,
! [X: tp__ty_2Erealax_2Ereal] : ( mem @ ( inj__ty_2Erealax_2Ereal @ X ) @ ty_2Erealax_2Ereal ) ).
thf(stp_iso_mem_ty_2Erealax_2Ereal,axiom,
! [X: $i] :
( ( mem @ X @ ty_2Erealax_2Ereal )
=> ( X
= ( inj__ty_2Erealax_2Ereal @ ( surj__ty_2Erealax_2Ereal @ X ) ) ) ) ).
thf(tp_c_2Erealax_2Ehreal__of__real,type,
c_2Erealax_2Ehreal__of__real: $i ).
thf(mem_c_2Erealax_2Ehreal__of__real,axiom,
mem @ c_2Erealax_2Ehreal__of__real @ ( arr @ ty_2Erealax_2Ereal @ ty_2Ehreal_2Ehreal ) ).
thf(stp_fo_c_2Erealax_2Ehreal__of__real,type,
fo__c_2Erealax_2Ehreal__of__real: tp__ty_2Erealax_2Ereal > tp__ty_2Ehreal_2Ehreal ).
thf(stp_eq_fo_c_2Erealax_2Ehreal__of__real,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Ehreal_2Ehreal @ ( fo__c_2Erealax_2Ehreal__of__real @ X0 ) )
= ( ap @ c_2Erealax_2Ehreal__of__real @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(stp_c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,type,
tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal: $tType ).
thf(stp_inj_c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,type,
inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal > $i ).
thf(stp_surj_c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,type,
surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal: $i > tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal ).
thf(stp_inj_surj_c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,axiom,
! [X: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,axiom,
! [X: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] : ( mem @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ X ) @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) ).
thf(stp_iso_mem_c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) )
=> ( X
= ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ X ) ) ) ) ).
thf(tp_c_2Erealax_2Ehreal__of__treal,type,
c_2Erealax_2Ehreal__of__treal: $i ).
thf(mem_c_2Erealax_2Ehreal__of__treal,axiom,
mem @ c_2Erealax_2Ehreal__of__treal @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ty_2Ehreal_2Ehreal ) ).
thf(stp_fo_c_2Erealax_2Ehreal__of__treal,type,
fo__c_2Erealax_2Ehreal__of__treal: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal > tp__ty_2Ehreal_2Ehreal ).
thf(stp_eq_fo_c_2Erealax_2Ehreal__of__treal,axiom,
! [X0: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( inj__ty_2Ehreal_2Ehreal @ ( fo__c_2Erealax_2Ehreal__of__treal @ X0 ) )
= ( ap @ c_2Erealax_2Ehreal__of__treal @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ X0 ) ) ) ).
thf(tp_c_2Erealax_2Einv,type,
c_2Erealax_2Einv: $i ).
thf(mem_c_2Erealax_2Einv,axiom,
mem @ c_2Erealax_2Einv @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Erealax_2Einv,type,
fo__c_2Erealax_2Einv: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Erealax_2Einv,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Erealax_2Einv @ X0 ) )
= ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Erealax_2Ereal__0,type,
c_2Erealax_2Ereal__0: $i ).
thf(mem_c_2Erealax_2Ereal__0,axiom,
mem @ c_2Erealax_2Ereal__0 @ ty_2Erealax_2Ereal ).
thf(stp_fo_c_2Erealax_2Ereal__0,type,
fo__c_2Erealax_2Ereal__0: tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Erealax_2Ereal__0,axiom,
( ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__0 )
= c_2Erealax_2Ereal__0 ) ).
thf(tp_c_2Erealax_2Ereal__1,type,
c_2Erealax_2Ereal__1: $i ).
thf(mem_c_2Erealax_2Ereal__1,axiom,
mem @ c_2Erealax_2Ereal__1 @ ty_2Erealax_2Ereal ).
thf(stp_fo_c_2Erealax_2Ereal__1,type,
fo__c_2Erealax_2Ereal__1: tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Erealax_2Ereal__1,axiom,
( ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__1 )
= c_2Erealax_2Ereal__1 ) ).
thf(tp_c_2Erealax_2Ereal__ABS,type,
c_2Erealax_2Ereal__ABS: $i ).
thf(mem_c_2Erealax_2Ereal__ABS,axiom,
mem @ c_2Erealax_2Ereal__ABS @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Erealax_2Ereal__ABS,type,
fo__c_2Erealax_2Ereal__ABS: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Erealax_2Ereal__ABS,axiom,
! [X0: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Erealax_2Ereal__ABS @ X0 ) )
= ( ap @ c_2Erealax_2Ereal__ABS @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ X0 ) ) ) ).
thf(tp_c_2Erealax_2Ereal__ABS__CLASS,type,
c_2Erealax_2Ereal__ABS__CLASS: $i ).
thf(mem_c_2Erealax_2Ereal__ABS__CLASS,axiom,
mem @ c_2Erealax_2Ereal__ABS__CLASS @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ bool ) @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Erealax_2Ereal__REP,type,
c_2Erealax_2Ereal__REP: $i ).
thf(mem_c_2Erealax_2Ereal__REP,axiom,
mem @ c_2Erealax_2Ereal__REP @ ( arr @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) ).
thf(tp_c_2Erealax_2Ereal__REP__CLASS,type,
c_2Erealax_2Ereal__REP__CLASS: $i ).
thf(mem_c_2Erealax_2Ereal__REP__CLASS,axiom,
mem @ c_2Erealax_2Ereal__REP__CLASS @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ bool ) ) ).
thf(tp_c_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: $i ).
thf(mem_c_2Erealax_2Ereal__add,axiom,
mem @ c_2Erealax_2Ereal__add @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ).
thf(stp_fo_c_2Erealax_2Ereal__add,type,
fo__c_2Erealax_2Ereal__add: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Erealax_2Ereal__add,axiom,
! [X0: tp__ty_2Erealax_2Ereal,X1: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Erealax_2Ereal__add @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) @ ( inj__ty_2Erealax_2Ereal @ X1 ) ) ) ).
thf(tp_c_2Erealax_2Ereal__lt,type,
c_2Erealax_2Ereal__lt: $i ).
thf(mem_c_2Erealax_2Ereal__lt,axiom,
mem @ c_2Erealax_2Ereal__lt @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Erealax_2Ereal__mul,type,
c_2Erealax_2Ereal__mul: $i ).
thf(mem_c_2Erealax_2Ereal__mul,axiom,
mem @ c_2Erealax_2Ereal__mul @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ).
thf(stp_fo_c_2Erealax_2Ereal__mul,type,
fo__c_2Erealax_2Ereal__mul: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Erealax_2Ereal__mul,axiom,
! [X0: tp__ty_2Erealax_2Ereal,X1: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Erealax_2Ereal__mul @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) @ ( inj__ty_2Erealax_2Ereal @ X1 ) ) ) ).
thf(tp_c_2Erealax_2Ereal__neg,type,
c_2Erealax_2Ereal__neg: $i ).
thf(mem_c_2Erealax_2Ereal__neg,axiom,
mem @ c_2Erealax_2Ereal__neg @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Erealax_2Ereal__neg,type,
fo__c_2Erealax_2Ereal__neg: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Erealax_2Ereal__neg,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Erealax_2Ereal__neg @ X0 ) )
= ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Erealax_2Ereal__of__hreal,type,
c_2Erealax_2Ereal__of__hreal: $i ).
thf(mem_c_2Erealax_2Ereal__of__hreal,axiom,
mem @ c_2Erealax_2Ereal__of__hreal @ ( arr @ ty_2Ehreal_2Ehreal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Erealax_2Ereal__of__hreal,type,
fo__c_2Erealax_2Ereal__of__hreal: tp__ty_2Ehreal_2Ehreal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Erealax_2Ereal__of__hreal,axiom,
! [X0: tp__ty_2Ehreal_2Ehreal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Erealax_2Ereal__of__hreal @ X0 ) )
= ( ap @ c_2Erealax_2Ereal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ X0 ) ) ) ).
thf(tp_c_2Erealax_2Etreal__0,type,
c_2Erealax_2Etreal__0: $i ).
thf(mem_c_2Erealax_2Etreal__0,axiom,
mem @ c_2Erealax_2Etreal__0 @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ).
thf(tp_c_2Erealax_2Etreal__1,type,
c_2Erealax_2Etreal__1: $i ).
thf(mem_c_2Erealax_2Etreal__1,axiom,
mem @ c_2Erealax_2Etreal__1 @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ).
thf(tp_c_2Erealax_2Etreal__add,type,
c_2Erealax_2Etreal__add: $i ).
thf(mem_c_2Erealax_2Etreal__add,axiom,
mem @ c_2Erealax_2Etreal__add @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) ) ).
thf(tp_c_2Erealax_2Etreal__eq,type,
c_2Erealax_2Etreal__eq: $i ).
thf(mem_c_2Erealax_2Etreal__eq,axiom,
mem @ c_2Erealax_2Etreal__eq @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ bool ) ) ).
thf(tp_c_2Erealax_2Etreal__inv,type,
c_2Erealax_2Etreal__inv: $i ).
thf(mem_c_2Erealax_2Etreal__inv,axiom,
mem @ c_2Erealax_2Etreal__inv @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) ).
thf(tp_c_2Erealax_2Etreal__lt,type,
c_2Erealax_2Etreal__lt: $i ).
thf(mem_c_2Erealax_2Etreal__lt,axiom,
mem @ c_2Erealax_2Etreal__lt @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ bool ) ) ).
thf(tp_c_2Erealax_2Etreal__mul,type,
c_2Erealax_2Etreal__mul: $i ).
thf(mem_c_2Erealax_2Etreal__mul,axiom,
mem @ c_2Erealax_2Etreal__mul @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) ) ).
thf(tp_c_2Erealax_2Etreal__neg,type,
c_2Erealax_2Etreal__neg: $i ).
thf(mem_c_2Erealax_2Etreal__neg,axiom,
mem @ c_2Erealax_2Etreal__neg @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) ).
thf(tp_c_2Erealax_2Etreal__of__hreal,type,
c_2Erealax_2Etreal__of__hreal: $i ).
thf(mem_c_2Erealax_2Etreal__of__hreal,axiom,
mem @ c_2Erealax_2Etreal__of__hreal @ ( arr @ ty_2Ehreal_2Ehreal @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) ).
thf(conj_thm_2Erealax_2EHREAL__RDISTRIB,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 @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( 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 @ V2z ) ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2z ) ) ) ) ) ).
thf(conj_thm_2Erealax_2EHREAL__EQ__ADDR,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(conj_thm_2Erealax_2EHREAL__EQ__ADDL,axiom,
! [V0x: tp__ty_2Ehreal_2Ehreal,V1y: tp__ty_2Ehreal_2Ehreal] :
( V0x
!= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) ) ) ).
thf(conj_thm_2Erealax_2EHREAL__EQ__LADD,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 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) )
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2z ) ) ) )
<=> ( V1y = V2z ) ) ).
thf(conj_thm_2Erealax_2EHREAL__LT__REFL,axiom,
! [V0x: tp__ty_2Ehreal_2Ehreal] :
~ ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) ) ).
thf(conj_thm_2Erealax_2EHREAL__LT__ADDL,axiom,
! [V0x: tp__ty_2Ehreal_2Ehreal,V1y: tp__ty_2Ehreal_2Ehreal] : ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) ) ) ).
thf(conj_thm_2Erealax_2EHREAL__LT__NE,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 ) ) ).
thf(conj_thm_2Erealax_2EHREAL__LT__ADDR,axiom,
! [V0x: tp__ty_2Ehreal_2Ehreal,V1y: tp__ty_2Ehreal_2Ehreal] :
~ ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) ) ).
thf(conj_thm_2Erealax_2EHREAL__LT__GT,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 @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) ) ) ).
thf(conj_thm_2Erealax_2EHREAL__LT__ADD2,axiom,
! [V0x1: tp__ty_2Ehreal_2Ehreal,V1x2: tp__ty_2Ehreal_2Ehreal,V2y1: tp__ty_2Ehreal_2Ehreal,V3y2: tp__ty_2Ehreal_2Ehreal] :
( ( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2y1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V1x2 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V3y2 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1x2 ) ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V2y1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V3y2 ) ) ) ) ) ).
thf(conj_thm_2Erealax_2EHREAL__LT__LADD,axiom,
! [V0x: tp__ty_2Ehreal_2Ehreal,V1y: tp__ty_2Ehreal_2Ehreal,V2z: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2z ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2z ) ) ) ) ).
thf(ax_thm_2Erealax_2Etreal__0,axiom,
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ c_2Erealax_2Etreal__0 )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 ) ) ) ) ).
thf(ax_thm_2Erealax_2Etreal__1,axiom,
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ c_2Erealax_2Etreal__1 )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 ) ) ) @ ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 ) ) ) ) ).
thf(ax_thm_2Erealax_2Etreal__neg,axiom,
! [V0x: tp__ty_2Ehreal_2Ehreal,V1y: tp__ty_2Ehreal_2Ehreal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ c_2Erealax_2Etreal__neg @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) ) ) ).
thf(ax_thm_2Erealax_2Etreal__add,axiom,
! [V0x1: tp__ty_2Ehreal_2Ehreal,V1y1: tp__ty_2Ehreal_2Ehreal,V2x2: tp__ty_2Ehreal_2Ehreal,V3y2: tp__ty_2Ehreal_2Ehreal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y1 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V2x2 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V3y2 ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2x2 ) ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V1y1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V3y2 ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Etreal__mul,axiom,
! [V0x1: tp__ty_2Ehreal_2Ehreal,V1y1: tp__ty_2Ehreal_2Ehreal,V2x2: tp__ty_2Ehreal_2Ehreal,V3y2: tp__ty_2Ehreal_2Ehreal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y1 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V2x2 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V3y2 ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2x2 ) ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V1y1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V3y2 ) ) ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V3y2 ) ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__mul @ ( inj__ty_2Ehreal_2Ehreal @ V1y1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V2x2 ) ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Etreal__lt,axiom,
! [V0x1: tp__ty_2Ehreal_2Ehreal,V1y1: tp__ty_2Ehreal_2Ehreal,V2x2: tp__ty_2Ehreal_2Ehreal,V3y2: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y1 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V2x2 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V3y2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V3y2 ) ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V2x2 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y1 ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Etreal__inv,axiom,
! [V0x: tp__ty_2Ehreal_2Ehreal,V1y: tp__ty_2Ehreal_2Ehreal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ c_2Erealax_2Etreal__inv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) ) @ c_2Erealax_2Etreal__0 ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( ap @ c_2Ehreal_2Ehreal__inv @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__sub @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) ) ) @ ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 ) ) ) @ ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( ap @ c_2Ehreal_2Ehreal__inv @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__sub @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) ) ) @ ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 ) ) ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Etreal__eq,axiom,
! [V0x1: tp__ty_2Ehreal_2Ehreal,V1y1: tp__ty_2Ehreal_2Ehreal,V2x2: tp__ty_2Ehreal_2Ehreal,V3y2: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y1 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V2x2 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V3y2 ) ) ) )
<=> ( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V3y2 ) ) )
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V2x2 ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y1 ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__EQ__REFL,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] : ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__EQ__SYM,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__EQ__TRANS,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2z ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2z ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__EQ__EQUIV,axiom,
! [V0p: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1q: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0p ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1q ) ) )
<=> ( ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0p ) )
= ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1q ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__EQ__AP,axiom,
! [V0p: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1q: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( V0p = V1q )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0p ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1q ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__10,axiom,
~ ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ c_2Erealax_2Etreal__1 ) @ c_2Erealax_2Etreal__0 ) ) ).
thf(conj_thm_2Erealax_2ETREAL__ADD__SYM,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__MUL__SYM,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__ADD__ASSOC,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2z ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__MUL__ASSOC,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2z ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__LDISTRIB,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2z ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__ADD__LID,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] : ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ c_2Erealax_2Etreal__0 ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__MUL__LID,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] : ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ c_2Erealax_2Etreal__1 ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__ADD__LINV,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] : ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( ap @ c_2Erealax_2Etreal__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) @ c_2Erealax_2Etreal__0 ) ) ).
thf(conj_thm_2Erealax_2ETREAL__INV__0,axiom,
p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ c_2Erealax_2Etreal__inv @ c_2Erealax_2Etreal__0 ) ) @ c_2Erealax_2Etreal__0 ) ).
thf(conj_thm_2Erealax_2ETREAL__MUL__LINV,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ~ ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ c_2Erealax_2Etreal__0 ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( ap @ c_2Erealax_2Etreal__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) @ c_2Erealax_2Etreal__1 ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__LT__TOTAL,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__LT__REFL,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
~ ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__LT__TRANS,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2z ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2z ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__LT__ADD,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2z ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2z ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__LT__MUL,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ c_2Erealax_2Etreal__0 ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ c_2Erealax_2Etreal__0 ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ c_2Erealax_2Etreal__0 ) @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Etreal__of__hreal,axiom,
! [V0x: tp__ty_2Ehreal_2Ehreal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ c_2Erealax_2Etreal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 ) ) ) @ ( inj__ty_2Ehreal_2Ehreal @ fo__c_2Ehreal_2Ehreal__1 ) ) ) ) ).
thf(ax_thm_2Erealax_2Ehreal__of__treal,axiom,
! [V0x: tp__ty_2Ehreal_2Ehreal,V1y: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ c_2Erealax_2Ehreal__of__treal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) ) )
= ( surj__ty_2Ehreal_2Ehreal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Ehreal_2Ehreal )
@ ( lam @ ty_2Ehreal_2Ehreal
@ ^ [V2d: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Ehreal_2Ehreal ) @ ( inj__ty_2Ehreal_2Ehreal @ V0x ) ) @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__add @ ( inj__ty_2Ehreal_2Ehreal @ V1y ) ) @ V2d ) ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__BIJ,axiom,
( ! [V0h: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ c_2Erealax_2Ehreal__of__treal @ ( ap @ c_2Erealax_2Etreal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V0h ) ) ) )
= V0h )
& ! [V1r: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ c_2Erealax_2Etreal__0 ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1r ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ c_2Erealax_2Etreal__of__hreal @ ( ap @ c_2Erealax_2Ehreal__of__treal @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1r ) ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1r ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__ISO,axiom,
! [V0h: tp__ty_2Ehreal_2Ehreal,V1i: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V0h ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1i ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( ap @ c_2Erealax_2Etreal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V0h ) ) ) @ ( ap @ c_2Erealax_2Etreal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V1i ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__BIJ__WELLDEF,axiom,
! [V0h: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1i: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0h ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1i ) ) )
=> ( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ c_2Erealax_2Ehreal__of__treal @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0h ) ) )
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ c_2Erealax_2Ehreal__of__treal @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1i ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__NEG__WELLDEF,axiom,
! [V0x1: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1x2: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ c_2Erealax_2Etreal__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) ) @ ( ap @ c_2Erealax_2Etreal__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__ADD__WELLDEFR,axiom,
! [V0x1: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1x2: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__ADD__WELLDEF,axiom,
! [V0x1: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1x2: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2y1: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V3y2: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V3y2 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y1 ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V3y2 ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__MUL__WELLDEFR,axiom,
! [V0x1: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1x2: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__MUL__WELLDEF,axiom,
! [V0x1: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1x2: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2y1: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V3y2: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V3y2 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y1 ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V3y2 ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__LT__WELLDEFR,axiom,
! [V0x1: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1x2: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2y: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__LT__WELLDEFL,axiom,
! [V0x: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1y1: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2y2: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y2 ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1y1 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y2 ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__LT__WELLDEF,axiom,
! [V0x1: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1x2: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V2y1: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V3y2: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V3y2 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2y1 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V3y2 ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ETREAL__INV__WELLDEF,axiom,
! [V0x1: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal,V1x2: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( ap @ c_2Erealax_2Etreal__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0x1 ) ) ) @ ( ap @ c_2Erealax_2Etreal__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V1x2 ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Ereal__TY__DEF,axiom,
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ bool ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ bool ) @ ty_2Erealax_2Ereal )
@ ( lam @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ bool )
@ ^ [V1c: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) )
@ ( lam @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal )
@ ^ [V2r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ V2r ) @ V2r ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ bool ) ) @ V1c ) @ ( ap @ c_2Erealax_2Etreal__eq @ V2r ) ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Erealax_2Ereal__bijections,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__ABS__CLASS @ ( ap @ c_2Erealax_2Ereal__REP__CLASS @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) )
= V0a )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ bool ) )
=> ( ( p
@ ( ap
@ ( lam @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ bool )
@ ^ [V2c: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) )
@ ( lam @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal )
@ ^ [V3r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ V3r ) @ V3r ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ bool ) ) @ V2c ) @ ( ap @ c_2Erealax_2Etreal__eq @ V3r ) ) ) ) ) )
@ V1r ) )
<=> ( ( ap @ c_2Erealax_2Ereal__REP__CLASS @ ( ap @ c_2Erealax_2Ereal__ABS__CLASS @ V1r ) )
= V1r ) ) ) ) ).
thf(conj_thm_2Erealax_2Ereal__ABS__REP__CLASS,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__ABS__CLASS @ ( ap @ c_2Erealax_2Ereal__REP__CLASS @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) )
= V0a )
& ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ bool ) )
=> ( ? [V2r: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2r ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2r ) ) )
& ( V1c
= ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V2r ) ) ) )
<=> ( ( ap @ c_2Erealax_2Ereal__REP__CLASS @ ( ap @ c_2Erealax_2Ereal__ABS__CLASS @ V1c ) )
= V1c ) ) ) ) ).
thf(ax_thm_2Erealax_2Ereal__REP__def,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ c_2Erealax_2Ereal__REP @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ ( ap @ ( c_2Emin_2E_40 @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) ) @ ( ap @ c_2Erealax_2Ereal__REP__CLASS @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Ereal__ABS__def,axiom,
! [V0r: tp__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__ABS @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0r ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__ABS__CLASS @ ( ap @ c_2Erealax_2Etreal__eq @ ( inj__c_ty_2Epair_2Eprod_ty_2Ehreal_2Ehreal_ty_2Ehreal_2Ehreal @ V0r ) ) ) ) ) ).
thf(conj_thm_2Erealax_2Ereal__QUOTIENT,axiom,
p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ ( ty_2Epair_2Eprod @ ty_2Ehreal_2Ehreal @ ty_2Ehreal_2Ehreal ) @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Etreal__eq ) @ c_2Erealax_2Ereal__ABS ) @ c_2Erealax_2Ereal__REP ) ).
thf(ax_thm_2Erealax_2Ereal__0,axiom,
( fo__c_2Erealax_2Ereal__0
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__ABS @ c_2Erealax_2Etreal__0 ) ) ) ).
thf(ax_thm_2Erealax_2Ereal__1,axiom,
( fo__c_2Erealax_2Ereal__1
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__ABS @ c_2Erealax_2Etreal__1 ) ) ) ).
thf(ax_thm_2Erealax_2Ereal__neg,axiom,
! [V0T1: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0T1 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__ABS @ ( ap @ c_2Erealax_2Etreal__neg @ ( ap @ c_2Erealax_2Ereal__REP @ ( inj__ty_2Erealax_2Ereal @ V0T1 ) ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Ereal__inv,axiom,
! [V0T1: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0T1 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__ABS @ ( ap @ c_2Erealax_2Etreal__inv @ ( ap @ c_2Erealax_2Ereal__REP @ ( inj__ty_2Erealax_2Ereal @ V0T1 ) ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Ereal__add,axiom,
! [V0T1: tp__ty_2Erealax_2Ereal,V1T2: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0T1 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1T2 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__ABS @ ( ap @ ( ap @ c_2Erealax_2Etreal__add @ ( ap @ c_2Erealax_2Ereal__REP @ ( inj__ty_2Erealax_2Ereal @ V0T1 ) ) ) @ ( ap @ c_2Erealax_2Ereal__REP @ ( inj__ty_2Erealax_2Ereal @ V1T2 ) ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Ereal__mul,axiom,
! [V0T1: tp__ty_2Erealax_2Ereal,V1T2: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0T1 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1T2 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__ABS @ ( ap @ ( ap @ c_2Erealax_2Etreal__mul @ ( ap @ c_2Erealax_2Ereal__REP @ ( inj__ty_2Erealax_2Ereal @ V0T1 ) ) ) @ ( ap @ c_2Erealax_2Ereal__REP @ ( inj__ty_2Erealax_2Ereal @ V1T2 ) ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Ereal__lt,axiom,
! [V0T1: tp__ty_2Erealax_2Ereal,V1T2: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0T1 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1T2 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Etreal__lt @ ( ap @ c_2Erealax_2Ereal__REP @ ( inj__ty_2Erealax_2Ereal @ V0T1 ) ) ) @ ( ap @ c_2Erealax_2Ereal__REP @ ( inj__ty_2Erealax_2Ereal @ V1T2 ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Ereal__of__hreal,axiom,
! [V0T1: tp__ty_2Ehreal_2Ehreal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V0T1 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__ABS @ ( ap @ c_2Erealax_2Etreal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V0T1 ) ) ) ) ) ).
thf(ax_thm_2Erealax_2Ehreal__of__real,axiom,
! [V0T1: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Ehreal_2Ehreal @ ( ap @ c_2Erealax_2Ehreal__of__real @ ( inj__ty_2Erealax_2Ereal @ V0T1 ) ) )
= ( surj__ty_2Ehreal_2Ehreal @ ( ap @ c_2Erealax_2Ehreal__of__treal @ ( ap @ c_2Erealax_2Ereal__REP @ ( inj__ty_2Erealax_2Ereal @ V0T1 ) ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__ISO__EQ,axiom,
! [V0h: tp__ty_2Ehreal_2Ehreal,V1i: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V0h ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V1i ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V0h ) ) ) @ ( ap @ c_2Erealax_2Ereal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V1i ) ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__POS,axiom,
! [V0X: tp__ty_2Ehreal_2Ehreal] : ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__0 ) ) @ ( ap @ c_2Erealax_2Ereal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V0X ) ) ) ) ).
thf(conj_thm_2Erealax_2ESUP__ALLPOS__LEMMA1,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1y: tp__ty_2Erealax_2Ereal] :
( ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
=> ( ? [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
<=> ? [V4X: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ V0P @ ( ap @ c_2Erealax_2Ereal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V4X ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Erealax_2Ereal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V4X ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ESUP__ALLPOS__LEMMA2,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1X: tp__ty_2Ehreal_2Ehreal] :
( ( p @ ( ap @ V0P @ ( ap @ c_2Erealax_2Ereal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V1X ) ) ) )
<=> ( p
@ ( ap
@ ( lam @ ty_2Ehreal_2Ehreal
@ ^ [V2h: $i] : ( ap @ V0P @ ( ap @ c_2Erealax_2Ereal__of__hreal @ V2h ) ) )
@ ( inj__ty_2Ehreal_2Ehreal @ V1X ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ESUP__ALLPOS__LEMMA3,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
& ? [V2x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ? [V3z: tp__ty_2Erealax_2Ereal] :
! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) ) )
=> ( ? [V5X: tp__ty_2Ehreal_2Ehreal] :
( p
@ ( ap
@ ( lam @ ty_2Ehreal_2Ehreal
@ ^ [V6h: $i] : ( ap @ V0P @ ( ap @ c_2Erealax_2Ereal__of__hreal @ V6h ) ) )
@ ( inj__ty_2Ehreal_2Ehreal @ V5X ) ) )
& ? [V7Y: tp__ty_2Ehreal_2Ehreal] :
! [V8X: tp__ty_2Ehreal_2Ehreal] :
( ( p
@ ( ap
@ ( lam @ ty_2Ehreal_2Ehreal
@ ^ [V9h: $i] : ( ap @ V0P @ ( ap @ c_2Erealax_2Ereal__of__hreal @ V9h ) ) )
@ ( inj__ty_2Ehreal_2Ehreal @ V8X ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ehreal_2Ehreal__lt @ ( inj__ty_2Ehreal_2Ehreal @ V8X ) ) @ ( inj__ty_2Ehreal_2Ehreal @ V7Y ) ) ) ) ) ) ) ).
thf(conj_thm_2Erealax_2ESUP__ALLPOS__LEMMA4,axiom,
! [V0y: tp__ty_2Erealax_2Ereal] :
( ~ ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) )
=> ! [V1x: tp__ty_2Ehreal_2Ehreal] : ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ c_2Erealax_2Ereal__of__hreal @ ( inj__ty_2Ehreal_2Ehreal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__SUP__ALLPOS,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
& ? [V2x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ? [V3z: tp__ty_2Erealax_2Ereal] :
! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) ) )
=> ? [V5s: tp__ty_2Erealax_2Ereal] :
! [V6y: tp__ty_2Erealax_2Ereal] :
( ? [V7x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V6y ) ) @ ( inj__ty_2Erealax_2Ereal @ V5s ) ) ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__10,axiom,
fo__c_2Erealax_2Ereal__1 != fo__c_2Erealax_2Ereal__0 ).
thf(conj_thm_2Erealax_2EREAL__ADD__SYM,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__MUL__SYM,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__ADD__ASSOC,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__MUL__ASSOC,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__LDISTRIB,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__ADD__LID,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= V0x ) ).
thf(conj_thm_2Erealax_2EREAL__MUL__LID,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__1 ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= V0x ) ).
thf(conj_thm_2Erealax_2EREAL__ADD__LINV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= fo__c_2Erealax_2Ereal__0 ) ).
thf(conj_thm_2Erealax_2EREAL__MUL__LINV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( V0x != fo__c_2Erealax_2Ereal__0 )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= fo__c_2Erealax_2Ereal__1 ) ) ).
thf(conj_thm_2Erealax_2EREAL__LT__TOTAL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( V0x = V1y )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__LT__REFL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
~ ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__LT__TRANS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__LT__IADD,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__LT__MUL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__0 ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Erealax_2EREAL__INV__0,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Erealax_2Ereal__0 ) ) )
= fo__c_2Erealax_2Ereal__0 ) ).
%------------------------------------------------------------------------------