ITP001 Axioms: ITP113^7.ax
%------------------------------------------------------------------------------
% File : ITP113^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : realax.ax [Gau19]
% : HL4113^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 145 ( 51 unt; 46 typ; 0 def)
% Number of atoms : 208 ( 63 equ; 14 cnn)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 947 ( 14 ~; 5 |; 22 &; 857 @)
% ( 14 <=>; 35 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg; 857 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 87 ( 87 >; 0 *; 0 +; 0 <<)
% Number of symbols : 45 ( 43 usr; 6 con; 0-5 aty)
% Number of variables : 230 ( 8 ^ 198 !; 13 ?; 230 :)
% ( 11 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Ehreal_2Ehreal,type,
tyop_2Ehreal_2Ehreal: $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Equotient_2EQUOTIENT,type,
c_2Equotient_2EQUOTIENT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > A_27b ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ehreal_2Ehreal__1,type,
c_2Ehreal_2Ehreal__1: tyop_2Ehreal_2Ehreal ).
thf(c_2Ehreal_2Ehreal__add,type,
c_2Ehreal_2Ehreal__add: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).
thf(c_2Ehreal_2Ehreal__inv,type,
c_2Ehreal_2Ehreal__inv: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).
thf(c_2Ehreal_2Ehreal__lt,type,
c_2Ehreal_2Ehreal__lt: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal > $o ).
thf(c_2Ehreal_2Ehreal__mul,type,
c_2Ehreal_2Ehreal__mul: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).
thf(c_2Erealax_2Ehreal__of__real,type,
c_2Erealax_2Ehreal__of__real: tyop_2Erealax_2Ereal > tyop_2Ehreal_2Ehreal ).
thf(c_2Erealax_2Ehreal__of__treal,type,
c_2Erealax_2Ehreal__of__treal: ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > tyop_2Ehreal_2Ehreal ).
thf(c_2Ehreal_2Ehreal__sub,type,
c_2Ehreal_2Ehreal__sub: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).
thf(c_2Erealax_2Einv,type,
c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__0,type,
c_2Erealax_2Ereal__0: tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__1,type,
c_2Erealax_2Ereal__1: tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__ABS,type,
c_2Erealax_2Ereal__ABS: ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__ABS__CLASS,type,
c_2Erealax_2Ereal__ABS__CLASS: ( ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > $o ) > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__REP,type,
c_2Erealax_2Ereal__REP: tyop_2Erealax_2Ereal > ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) ).
thf(c_2Erealax_2Ereal__REP__CLASS,type,
c_2Erealax_2Ereal__REP__CLASS: tyop_2Erealax_2Ereal > ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > $o ).
thf(c_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__lt,type,
c_2Erealax_2Ereal__lt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Ereal__mul,type,
c_2Erealax_2Ereal__mul: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__neg,type,
c_2Erealax_2Ereal__neg: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__of__hreal,type,
c_2Erealax_2Ereal__of__hreal: tyop_2Ehreal_2Ehreal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Etreal__0,type,
c_2Erealax_2Etreal__0: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ).
thf(c_2Erealax_2Etreal__1,type,
c_2Erealax_2Etreal__1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ).
thf(c_2Erealax_2Etreal__add,type,
c_2Erealax_2Etreal__add: ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) ).
thf(c_2Erealax_2Etreal__eq,type,
c_2Erealax_2Etreal__eq: ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > $o ).
thf(c_2Erealax_2Etreal__inv,type,
c_2Erealax_2Etreal__inv: ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) ).
thf(c_2Erealax_2Etreal__lt,type,
c_2Erealax_2Etreal__lt: ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > $o ).
thf(c_2Erealax_2Etreal__mul,type,
c_2Erealax_2Etreal__mul: ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) ).
thf(c_2Erealax_2Etreal__neg,type,
c_2Erealax_2Etreal__neg: ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) ).
thf(c_2Erealax_2Etreal__of__hreal,type,
c_2Erealax_2Etreal__of__hreal: tyop_2Ehreal_2Ehreal > ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Erealax_2Etreal__0,axiom,
( c_2Erealax_2Etreal__0
= ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ c_2Ehreal_2Ehreal__1 @ c_2Ehreal_2Ehreal__1 ) ) ).
thf(thm_2Erealax_2Etreal__1,axiom,
( c_2Erealax_2Etreal__1
= ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ ( c_2Ehreal_2Ehreal__add @ c_2Ehreal_2Ehreal__1 @ c_2Ehreal_2Ehreal__1 ) @ c_2Ehreal_2Ehreal__1 ) ) ).
thf(thm_2Erealax_2Etreal__neg,axiom,
! [V0x: tyop_2Ehreal_2Ehreal,V1y: tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__neg @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ V0x @ V1y ) )
= ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ V1y @ V0x ) ) ).
thf(thm_2Erealax_2Etreal__add,axiom,
! [V0x1: tyop_2Ehreal_2Ehreal,V1y1: tyop_2Ehreal_2Ehreal,V2x2: tyop_2Ehreal_2Ehreal,V3y2: tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__add @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ V0x1 @ V1y1 ) @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ V2x2 @ V3y2 ) )
= ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ ( c_2Ehreal_2Ehreal__add @ V0x1 @ V2x2 ) @ ( c_2Ehreal_2Ehreal__add @ V1y1 @ V3y2 ) ) ) ).
thf(thm_2Erealax_2Etreal__mul,axiom,
! [V0x1: tyop_2Ehreal_2Ehreal,V1y1: tyop_2Ehreal_2Ehreal,V2x2: tyop_2Ehreal_2Ehreal,V3y2: tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__mul @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ V0x1 @ V1y1 ) @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ V2x2 @ V3y2 ) )
= ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ ( c_2Ehreal_2Ehreal__add @ ( c_2Ehreal_2Ehreal__mul @ V0x1 @ V2x2 ) @ ( c_2Ehreal_2Ehreal__mul @ V1y1 @ V3y2 ) ) @ ( c_2Ehreal_2Ehreal__add @ ( c_2Ehreal_2Ehreal__mul @ V0x1 @ V3y2 ) @ ( c_2Ehreal_2Ehreal__mul @ V1y1 @ V2x2 ) ) ) ) ).
thf(thm_2Erealax_2Etreal__lt,axiom,
! [V0x1: tyop_2Ehreal_2Ehreal,V1y1: tyop_2Ehreal_2Ehreal,V2x2: tyop_2Ehreal_2Ehreal,V3y2: tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__lt @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ V0x1 @ V1y1 ) @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ V2x2 @ V3y2 ) )
= ( c_2Ehreal_2Ehreal__lt @ ( c_2Ehreal_2Ehreal__add @ V0x1 @ V3y2 ) @ ( c_2Ehreal_2Ehreal__add @ V2x2 @ V1y1 ) ) ) ).
thf(thm_2Erealax_2Etreal__inv,axiom,
! [V0x: tyop_2Ehreal_2Ehreal,V1y: tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__inv @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ V0x @ V1y ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) @ ( c_2Emin_2E_3D @ tyop_2Ehreal_2Ehreal @ V0x @ V1y ) @ c_2Erealax_2Etreal__0 @ ( c_2Ebool_2ECOND @ ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) @ ( c_2Ehreal_2Ehreal__lt @ V1y @ V0x ) @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ ( c_2Ehreal_2Ehreal__add @ ( c_2Ehreal_2Ehreal__inv @ ( c_2Ehreal_2Ehreal__sub @ V0x @ V1y ) ) @ c_2Ehreal_2Ehreal__1 ) @ c_2Ehreal_2Ehreal__1 ) @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ c_2Ehreal_2Ehreal__1 @ ( c_2Ehreal_2Ehreal__add @ ( c_2Ehreal_2Ehreal__inv @ ( c_2Ehreal_2Ehreal__sub @ V1y @ V0x ) ) @ c_2Ehreal_2Ehreal__1 ) ) ) ) ) ).
thf(thm_2Erealax_2Etreal__eq,axiom,
! [V0x1: tyop_2Ehreal_2Ehreal,V1y1: tyop_2Ehreal_2Ehreal,V2x2: tyop_2Ehreal_2Ehreal,V3y2: tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__eq @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ V0x1 @ V1y1 ) @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ V2x2 @ V3y2 ) )
<=> ( ( c_2Ehreal_2Ehreal__add @ V0x1 @ V3y2 )
= ( c_2Ehreal_2Ehreal__add @ V2x2 @ V1y1 ) ) ) ).
thf(thm_2Erealax_2Etreal__of__hreal,axiom,
! [V0x: tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__of__hreal @ V0x )
= ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ ( c_2Ehreal_2Ehreal__add @ V0x @ c_2Ehreal_2Ehreal__1 ) @ c_2Ehreal_2Ehreal__1 ) ) ).
thf(thm_2Erealax_2Ehreal__of__treal,axiom,
! [V0x: tyop_2Ehreal_2Ehreal,V1y: tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Ehreal__of__treal @ ( c_2Epair_2E_2C @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal @ V0x @ V1y ) )
= ( c_2Emin_2E_40 @ tyop_2Ehreal_2Ehreal
@ ^ [V2d: tyop_2Ehreal_2Ehreal] : ( c_2Emin_2E_3D @ tyop_2Ehreal_2Ehreal @ V0x @ ( c_2Ehreal_2Ehreal__add @ V1y @ V2d ) ) ) ) ).
thf(thm_2Erealax_2Ereal__TY__DEF,axiom,
? [V0rep: tyop_2Erealax_2Ereal > ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > $o] :
( c_2Ebool_2ETYPE__DEFINITION @ ( ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > $o ) @ tyop_2Erealax_2Ereal
@ ^ [V1c: ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > $o] :
( c_2Ebool_2E_3F @ ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal )
@ ^ [V2r: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] : ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Etreal__eq @ V2r @ V2r ) @ ( c_2Emin_2E_3D @ ( ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > $o ) @ V1c @ ( c_2Erealax_2Etreal__eq @ V2r ) ) ) )
@ V0rep ) ).
thf(thm_2Erealax_2Ereal__bijections,axiom,
( ! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__ABS__CLASS @ ( c_2Erealax_2Ereal__REP__CLASS @ V0a ) )
= V0a )
& ! [V1r: ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > $o] :
( ( ^ [V2c: ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > $o] :
( c_2Ebool_2E_3F @ ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal )
@ ^ [V3r: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] : ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Etreal__eq @ V3r @ V3r ) @ ( c_2Emin_2E_3D @ ( ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > $o ) @ V2c @ ( c_2Erealax_2Etreal__eq @ V3r ) ) ) )
@ V1r )
<=> ( ( c_2Erealax_2Ereal__REP__CLASS @ ( c_2Erealax_2Ereal__ABS__CLASS @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Erealax_2Ereal__REP__def,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__REP @ V0a )
= ( c_2Emin_2E_40 @ ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) @ ( c_2Erealax_2Ereal__REP__CLASS @ V0a ) ) ) ).
thf(thm_2Erealax_2Ereal__ABS__def,axiom,
! [V0r: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Ereal__ABS @ V0r )
= ( c_2Erealax_2Ereal__ABS__CLASS @ ( c_2Erealax_2Etreal__eq @ V0r ) ) ) ).
thf(thm_2Erealax_2Ereal__0,axiom,
( c_2Erealax_2Ereal__0
= ( c_2Erealax_2Ereal__ABS @ c_2Erealax_2Etreal__0 ) ) ).
thf(thm_2Erealax_2Ereal__1,axiom,
( c_2Erealax_2Ereal__1
= ( c_2Erealax_2Ereal__ABS @ c_2Erealax_2Etreal__1 ) ) ).
thf(thm_2Erealax_2Ereal__neg,axiom,
! [V0T1: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg @ V0T1 )
= ( c_2Erealax_2Ereal__ABS @ ( c_2Erealax_2Etreal__neg @ ( c_2Erealax_2Ereal__REP @ V0T1 ) ) ) ) ).
thf(thm_2Erealax_2Ereal__inv,axiom,
! [V0T1: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Einv @ V0T1 )
= ( c_2Erealax_2Ereal__ABS @ ( c_2Erealax_2Etreal__inv @ ( c_2Erealax_2Ereal__REP @ V0T1 ) ) ) ) ).
thf(thm_2Erealax_2Ereal__add,axiom,
! [V0T1: tyop_2Erealax_2Ereal,V1T2: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V0T1 @ V1T2 )
= ( c_2Erealax_2Ereal__ABS @ ( c_2Erealax_2Etreal__add @ ( c_2Erealax_2Ereal__REP @ V0T1 ) @ ( c_2Erealax_2Ereal__REP @ V1T2 ) ) ) ) ).
thf(thm_2Erealax_2Ereal__mul,axiom,
! [V0T1: tyop_2Erealax_2Ereal,V1T2: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0T1 @ V1T2 )
= ( c_2Erealax_2Ereal__ABS @ ( c_2Erealax_2Etreal__mul @ ( c_2Erealax_2Ereal__REP @ V0T1 ) @ ( c_2Erealax_2Ereal__REP @ V1T2 ) ) ) ) ).
thf(thm_2Erealax_2Ereal__lt,axiom,
! [V0T1: tyop_2Erealax_2Ereal,V1T2: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0T1 @ V1T2 )
= ( c_2Erealax_2Etreal__lt @ ( c_2Erealax_2Ereal__REP @ V0T1 ) @ ( c_2Erealax_2Ereal__REP @ V1T2 ) ) ) ).
thf(thm_2Erealax_2Ereal__of__hreal,axiom,
! [V0T1: tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Ereal__of__hreal @ V0T1 )
= ( c_2Erealax_2Ereal__ABS @ ( c_2Erealax_2Etreal__of__hreal @ V0T1 ) ) ) ).
thf(thm_2Erealax_2Ehreal__of__real,axiom,
! [V0T1: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ehreal__of__real @ V0T1 )
= ( c_2Erealax_2Ehreal__of__treal @ ( c_2Erealax_2Ereal__REP @ V0T1 ) ) ) ).
thf(thm_2Erealax_2EHREAL__RDISTRIB,axiom,
! [V0x: tyop_2Ehreal_2Ehreal,V1y: tyop_2Ehreal_2Ehreal,V2z: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__mul @ ( c_2Ehreal_2Ehreal__add @ V0x @ V1y ) @ V2z )
= ( c_2Ehreal_2Ehreal__add @ ( c_2Ehreal_2Ehreal__mul @ V0x @ V2z ) @ ( c_2Ehreal_2Ehreal__mul @ V1y @ V2z ) ) ) ).
thf(thm_2Erealax_2EHREAL__EQ__ADDR,axiom,
! [V0x: tyop_2Ehreal_2Ehreal,V1y: tyop_2Ehreal_2Ehreal] :
( (~)
@ ( ( c_2Ehreal_2Ehreal__add @ V0x @ V1y )
= V0x ) ) ).
thf(thm_2Erealax_2EHREAL__EQ__ADDL,axiom,
! [V0x: tyop_2Ehreal_2Ehreal,V1y: tyop_2Ehreal_2Ehreal] :
( (~)
@ ( V0x
= ( c_2Ehreal_2Ehreal__add @ V0x @ V1y ) ) ) ).
thf(thm_2Erealax_2EHREAL__EQ__LADD,axiom,
! [V0x: tyop_2Ehreal_2Ehreal,V1y: tyop_2Ehreal_2Ehreal,V2z: tyop_2Ehreal_2Ehreal] :
( ( ( c_2Ehreal_2Ehreal__add @ V0x @ V1y )
= ( c_2Ehreal_2Ehreal__add @ V0x @ V2z ) )
<=> ( V1y = V2z ) ) ).
thf(thm_2Erealax_2EHREAL__LT__REFL,axiom,
! [V0x: tyop_2Ehreal_2Ehreal] : ( (~) @ ( c_2Ehreal_2Ehreal__lt @ V0x @ V0x ) ) ).
thf(thm_2Erealax_2EHREAL__LT__ADDL,axiom,
! [V0x: tyop_2Ehreal_2Ehreal,V1y: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__lt @ V0x @ ( c_2Ehreal_2Ehreal__add @ V0x @ V1y ) ) ).
thf(thm_2Erealax_2EHREAL__LT__NE,axiom,
! [V0x: tyop_2Ehreal_2Ehreal,V1y: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__lt @ V0x @ V1y )
=> ( (~) @ ( V0x = V1y ) ) ) ).
thf(thm_2Erealax_2EHREAL__LT__ADDR,axiom,
! [V0x: tyop_2Ehreal_2Ehreal,V1y: tyop_2Ehreal_2Ehreal] : ( (~) @ ( c_2Ehreal_2Ehreal__lt @ ( c_2Ehreal_2Ehreal__add @ V0x @ V1y ) @ V0x ) ) ).
thf(thm_2Erealax_2EHREAL__LT__GT,axiom,
! [V0x: tyop_2Ehreal_2Ehreal,V1y: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__lt @ V0x @ V1y )
=> ( (~) @ ( c_2Ehreal_2Ehreal__lt @ V1y @ V0x ) ) ) ).
thf(thm_2Erealax_2EHREAL__LT__ADD2,axiom,
! [V0x1: tyop_2Ehreal_2Ehreal,V1x2: tyop_2Ehreal_2Ehreal,V2y1: tyop_2Ehreal_2Ehreal,V3y2: tyop_2Ehreal_2Ehreal] :
( ( ( c_2Ehreal_2Ehreal__lt @ V0x1 @ V2y1 )
& ( c_2Ehreal_2Ehreal__lt @ V1x2 @ V3y2 ) )
=> ( c_2Ehreal_2Ehreal__lt @ ( c_2Ehreal_2Ehreal__add @ V0x1 @ V1x2 ) @ ( c_2Ehreal_2Ehreal__add @ V2y1 @ V3y2 ) ) ) ).
thf(thm_2Erealax_2EHREAL__LT__LADD,axiom,
! [V0x: tyop_2Ehreal_2Ehreal,V1y: tyop_2Ehreal_2Ehreal,V2z: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__lt @ ( c_2Ehreal_2Ehreal__add @ V0x @ V1y ) @ ( c_2Ehreal_2Ehreal__add @ V0x @ V2z ) )
= ( c_2Ehreal_2Ehreal__lt @ V1y @ V2z ) ) ).
thf(thm_2Erealax_2ETREAL__EQ__REFL,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Etreal__eq @ V0x @ V0x ) ).
thf(thm_2Erealax_2ETREAL__EQ__SYM,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__eq @ V0x @ V1y )
= ( c_2Erealax_2Etreal__eq @ V1y @ V0x ) ) ).
thf(thm_2Erealax_2ETREAL__EQ__TRANS,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2z: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( ( c_2Erealax_2Etreal__eq @ V0x @ V1y )
& ( c_2Erealax_2Etreal__eq @ V1y @ V2z ) )
=> ( c_2Erealax_2Etreal__eq @ V0x @ V2z ) ) ).
thf(thm_2Erealax_2ETREAL__EQ__EQUIV,axiom,
! [V0p: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1q: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__eq @ V0p @ V1q )
<=> ( ( c_2Erealax_2Etreal__eq @ V0p )
= ( c_2Erealax_2Etreal__eq @ V1q ) ) ) ).
thf(thm_2Erealax_2ETREAL__EQ__AP,axiom,
! [V0p: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1q: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( V0p = V1q )
=> ( c_2Erealax_2Etreal__eq @ V0p @ V1q ) ) ).
thf(thm_2Erealax_2ETREAL__10,axiom,
(~) @ ( c_2Erealax_2Etreal__eq @ c_2Erealax_2Etreal__1 @ c_2Erealax_2Etreal__0 ) ).
thf(thm_2Erealax_2ETREAL__ADD__SYM,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__add @ V0x @ V1y )
= ( c_2Erealax_2Etreal__add @ V1y @ V0x ) ) ).
thf(thm_2Erealax_2ETREAL__MUL__SYM,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__mul @ V0x @ V1y )
= ( c_2Erealax_2Etreal__mul @ V1y @ V0x ) ) ).
thf(thm_2Erealax_2ETREAL__ADD__ASSOC,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2z: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__add @ V0x @ ( c_2Erealax_2Etreal__add @ V1y @ V2z ) )
= ( c_2Erealax_2Etreal__add @ ( c_2Erealax_2Etreal__add @ V0x @ V1y ) @ V2z ) ) ).
thf(thm_2Erealax_2ETREAL__MUL__ASSOC,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2z: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__mul @ V0x @ ( c_2Erealax_2Etreal__mul @ V1y @ V2z ) )
= ( c_2Erealax_2Etreal__mul @ ( c_2Erealax_2Etreal__mul @ V0x @ V1y ) @ V2z ) ) ).
thf(thm_2Erealax_2ETREAL__LDISTRIB,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2z: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__mul @ V0x @ ( c_2Erealax_2Etreal__add @ V1y @ V2z ) )
= ( c_2Erealax_2Etreal__add @ ( c_2Erealax_2Etreal__mul @ V0x @ V1y ) @ ( c_2Erealax_2Etreal__mul @ V0x @ V2z ) ) ) ).
thf(thm_2Erealax_2ETREAL__ADD__LID,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Etreal__eq @ ( c_2Erealax_2Etreal__add @ c_2Erealax_2Etreal__0 @ V0x ) @ V0x ) ).
thf(thm_2Erealax_2ETREAL__MUL__LID,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Etreal__eq @ ( c_2Erealax_2Etreal__mul @ c_2Erealax_2Etreal__1 @ V0x ) @ V0x ) ).
thf(thm_2Erealax_2ETREAL__ADD__LINV,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Etreal__eq @ ( c_2Erealax_2Etreal__add @ ( c_2Erealax_2Etreal__neg @ V0x ) @ V0x ) @ c_2Erealax_2Etreal__0 ) ).
thf(thm_2Erealax_2ETREAL__INV__0,axiom,
c_2Erealax_2Etreal__eq @ ( c_2Erealax_2Etreal__inv @ c_2Erealax_2Etreal__0 ) @ c_2Erealax_2Etreal__0 ).
thf(thm_2Erealax_2ETREAL__MUL__LINV,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( (~) @ ( c_2Erealax_2Etreal__eq @ V0x @ c_2Erealax_2Etreal__0 ) )
=> ( c_2Erealax_2Etreal__eq @ ( c_2Erealax_2Etreal__mul @ ( c_2Erealax_2Etreal__inv @ V0x ) @ V0x ) @ c_2Erealax_2Etreal__1 ) ) ).
thf(thm_2Erealax_2ETREAL__LT__TOTAL,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__eq @ V0x @ V1y )
| ( c_2Erealax_2Etreal__lt @ V0x @ V1y )
| ( c_2Erealax_2Etreal__lt @ V1y @ V0x ) ) ).
thf(thm_2Erealax_2ETREAL__LT__REFL,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] : ( (~) @ ( c_2Erealax_2Etreal__lt @ V0x @ V0x ) ) ).
thf(thm_2Erealax_2ETREAL__LT__TRANS,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2z: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( ( c_2Erealax_2Etreal__lt @ V0x @ V1y )
& ( c_2Erealax_2Etreal__lt @ V1y @ V2z ) )
=> ( c_2Erealax_2Etreal__lt @ V0x @ V2z ) ) ).
thf(thm_2Erealax_2ETREAL__LT__ADD,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2z: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__lt @ V1y @ V2z )
=> ( c_2Erealax_2Etreal__lt @ ( c_2Erealax_2Etreal__add @ V0x @ V1y ) @ ( c_2Erealax_2Etreal__add @ V0x @ V2z ) ) ) ).
thf(thm_2Erealax_2ETREAL__LT__MUL,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( ( c_2Erealax_2Etreal__lt @ c_2Erealax_2Etreal__0 @ V0x )
& ( c_2Erealax_2Etreal__lt @ c_2Erealax_2Etreal__0 @ V1y ) )
=> ( c_2Erealax_2Etreal__lt @ c_2Erealax_2Etreal__0 @ ( c_2Erealax_2Etreal__mul @ V0x @ V1y ) ) ) ).
thf(thm_2Erealax_2ETREAL__BIJ,axiom,
( ! [V0h: tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Ehreal__of__treal @ ( c_2Erealax_2Etreal__of__hreal @ V0h ) )
= V0h )
& ! [V1r: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__lt @ c_2Erealax_2Etreal__0 @ V1r )
= ( c_2Erealax_2Etreal__eq @ ( c_2Erealax_2Etreal__of__hreal @ ( c_2Erealax_2Ehreal__of__treal @ V1r ) ) @ V1r ) ) ) ).
thf(thm_2Erealax_2ETREAL__ISO,axiom,
! [V0h: tyop_2Ehreal_2Ehreal,V1i: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__lt @ V0h @ V1i )
=> ( c_2Erealax_2Etreal__lt @ ( c_2Erealax_2Etreal__of__hreal @ V0h ) @ ( c_2Erealax_2Etreal__of__hreal @ V1i ) ) ) ).
thf(thm_2Erealax_2ETREAL__BIJ__WELLDEF,axiom,
! [V0h: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1i: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__eq @ V0h @ V1i )
=> ( ( c_2Erealax_2Ehreal__of__treal @ V0h )
= ( c_2Erealax_2Ehreal__of__treal @ V1i ) ) ) ).
thf(thm_2Erealax_2ETREAL__NEG__WELLDEF,axiom,
! [V0x1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1x2: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__eq @ V0x1 @ V1x2 )
=> ( c_2Erealax_2Etreal__eq @ ( c_2Erealax_2Etreal__neg @ V0x1 ) @ ( c_2Erealax_2Etreal__neg @ V1x2 ) ) ) ).
thf(thm_2Erealax_2ETREAL__ADD__WELLDEFR,axiom,
! [V0x1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1x2: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__eq @ V0x1 @ V1x2 )
=> ( c_2Erealax_2Etreal__eq @ ( c_2Erealax_2Etreal__add @ V0x1 @ V2y ) @ ( c_2Erealax_2Etreal__add @ V1x2 @ V2y ) ) ) ).
thf(thm_2Erealax_2ETREAL__ADD__WELLDEF,axiom,
! [V0x1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1x2: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2y1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V3y2: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( ( c_2Erealax_2Etreal__eq @ V0x1 @ V1x2 )
& ( c_2Erealax_2Etreal__eq @ V2y1 @ V3y2 ) )
=> ( c_2Erealax_2Etreal__eq @ ( c_2Erealax_2Etreal__add @ V0x1 @ V2y1 ) @ ( c_2Erealax_2Etreal__add @ V1x2 @ V3y2 ) ) ) ).
thf(thm_2Erealax_2ETREAL__MUL__WELLDEFR,axiom,
! [V0x1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1x2: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__eq @ V0x1 @ V1x2 )
=> ( c_2Erealax_2Etreal__eq @ ( c_2Erealax_2Etreal__mul @ V0x1 @ V2y ) @ ( c_2Erealax_2Etreal__mul @ V1x2 @ V2y ) ) ) ).
thf(thm_2Erealax_2ETREAL__MUL__WELLDEF,axiom,
! [V0x1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1x2: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2y1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V3y2: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( ( c_2Erealax_2Etreal__eq @ V0x1 @ V1x2 )
& ( c_2Erealax_2Etreal__eq @ V2y1 @ V3y2 ) )
=> ( c_2Erealax_2Etreal__eq @ ( c_2Erealax_2Etreal__mul @ V0x1 @ V2y1 ) @ ( c_2Erealax_2Etreal__mul @ V1x2 @ V3y2 ) ) ) ).
thf(thm_2Erealax_2ETREAL__LT__WELLDEFR,axiom,
! [V0x1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1x2: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2y: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__eq @ V0x1 @ V1x2 )
=> ( ( c_2Erealax_2Etreal__lt @ V0x1 @ V2y )
= ( c_2Erealax_2Etreal__lt @ V1x2 @ V2y ) ) ) ).
thf(thm_2Erealax_2ETREAL__LT__WELLDEFL,axiom,
! [V0x: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1y1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2y2: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__eq @ V1y1 @ V2y2 )
=> ( ( c_2Erealax_2Etreal__lt @ V0x @ V1y1 )
= ( c_2Erealax_2Etreal__lt @ V0x @ V2y2 ) ) ) ).
thf(thm_2Erealax_2ETREAL__LT__WELLDEF,axiom,
! [V0x1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1x2: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V2y1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V3y2: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( ( c_2Erealax_2Etreal__eq @ V0x1 @ V1x2 )
& ( c_2Erealax_2Etreal__eq @ V2y1 @ V3y2 ) )
=> ( ( c_2Erealax_2Etreal__lt @ V0x1 @ V2y1 )
= ( c_2Erealax_2Etreal__lt @ V1x2 @ V3y2 ) ) ) ).
thf(thm_2Erealax_2ETREAL__INV__WELLDEF,axiom,
! [V0x1: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal,V1x2: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__eq @ V0x1 @ V1x2 )
=> ( c_2Erealax_2Etreal__eq @ ( c_2Erealax_2Etreal__inv @ V0x1 ) @ ( c_2Erealax_2Etreal__inv @ V1x2 ) ) ) ).
thf(thm_2Erealax_2Ereal__ABS__REP__CLASS,axiom,
( ! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__ABS__CLASS @ ( c_2Erealax_2Ereal__REP__CLASS @ V0a ) )
= V0a )
& ! [V1c: ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) > $o] :
( ? [V2r: tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal] :
( ( c_2Erealax_2Etreal__eq @ V2r @ V2r )
& ( V1c
= ( c_2Erealax_2Etreal__eq @ V2r ) ) )
<=> ( ( c_2Erealax_2Ereal__REP__CLASS @ ( c_2Erealax_2Ereal__ABS__CLASS @ V1c ) )
= V1c ) ) ) ).
thf(thm_2Erealax_2Ereal__QUOTIENT,axiom,
c_2Equotient_2EQUOTIENT @ ( tyop_2Epair_2Eprod @ tyop_2Ehreal_2Ehreal @ tyop_2Ehreal_2Ehreal ) @ tyop_2Erealax_2Ereal @ c_2Erealax_2Etreal__eq @ c_2Erealax_2Ereal__ABS @ c_2Erealax_2Ereal__REP ).
thf(thm_2Erealax_2EREAL__ISO__EQ,axiom,
! [V0h: tyop_2Ehreal_2Ehreal,V1i: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__lt @ V0h @ V1i )
= ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__of__hreal @ V0h ) @ ( c_2Erealax_2Ereal__of__hreal @ V1i ) ) ) ).
thf(thm_2Erealax_2EREAL__POS,axiom,
! [V0X: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Ereal__lt @ c_2Erealax_2Ereal__0 @ ( c_2Erealax_2Ereal__of__hreal @ V0X ) ) ).
thf(thm_2Erealax_2ESUP__ALLPOS__LEMMA1,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1y: tyop_2Erealax_2Ereal] :
( ! [V2x: tyop_2Erealax_2Ereal] :
( ( V0P @ V2x )
=> ( c_2Erealax_2Ereal__lt @ c_2Erealax_2Ereal__0 @ V2x ) )
=> ( ? [V3x: tyop_2Erealax_2Ereal] :
( ( V0P @ V3x )
& ( c_2Erealax_2Ereal__lt @ V1y @ V3x ) )
<=> ? [V4X: tyop_2Ehreal_2Ehreal] :
( ( V0P @ ( c_2Erealax_2Ereal__of__hreal @ V4X ) )
& ( c_2Erealax_2Ereal__lt @ V1y @ ( c_2Erealax_2Ereal__of__hreal @ V4X ) ) ) ) ) ).
thf(thm_2Erealax_2ESUP__ALLPOS__LEMMA2,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o,V1X: tyop_2Ehreal_2Ehreal] :
( ( V0P @ ( c_2Erealax_2Ereal__of__hreal @ V1X ) )
= ( ^ [V2h: tyop_2Ehreal_2Ehreal] : ( V0P @ ( c_2Erealax_2Ereal__of__hreal @ V2h ) )
@ V1X ) ) ).
thf(thm_2Erealax_2ESUP__ALLPOS__LEMMA3,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ! [V1x: tyop_2Erealax_2Ereal] :
( ( V0P @ V1x )
=> ( c_2Erealax_2Ereal__lt @ c_2Erealax_2Ereal__0 @ V1x ) )
& ? [V2x: tyop_2Erealax_2Ereal] : ( V0P @ V2x )
& ? [V3z: tyop_2Erealax_2Ereal] :
! [V4x: tyop_2Erealax_2Ereal] :
( ( V0P @ V4x )
=> ( c_2Erealax_2Ereal__lt @ V4x @ V3z ) ) )
=> ( ? [V5X: tyop_2Ehreal_2Ehreal] :
( ^ [V6h: tyop_2Ehreal_2Ehreal] : ( V0P @ ( c_2Erealax_2Ereal__of__hreal @ V6h ) )
@ V5X )
& ? [V7Y: tyop_2Ehreal_2Ehreal] :
! [V8X: tyop_2Ehreal_2Ehreal] :
( ( ^ [V9h: tyop_2Ehreal_2Ehreal] : ( V0P @ ( c_2Erealax_2Ereal__of__hreal @ V9h ) )
@ V8X )
=> ( c_2Ehreal_2Ehreal__lt @ V8X @ V7Y ) ) ) ) ).
thf(thm_2Erealax_2ESUP__ALLPOS__LEMMA4,axiom,
! [V0y: tyop_2Erealax_2Ereal] :
( ( (~) @ ( c_2Erealax_2Ereal__lt @ c_2Erealax_2Ereal__0 @ V0y ) )
=> ! [V1x: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Ereal__lt @ V0y @ ( c_2Erealax_2Ereal__of__hreal @ V1x ) ) ) ).
thf(thm_2Erealax_2EREAL__SUP__ALLPOS,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ! [V1x: tyop_2Erealax_2Ereal] :
( ( V0P @ V1x )
=> ( c_2Erealax_2Ereal__lt @ c_2Erealax_2Ereal__0 @ V1x ) )
& ? [V2x: tyop_2Erealax_2Ereal] : ( V0P @ V2x )
& ? [V3z: tyop_2Erealax_2Ereal] :
! [V4x: tyop_2Erealax_2Ereal] :
( ( V0P @ V4x )
=> ( c_2Erealax_2Ereal__lt @ V4x @ V3z ) ) )
=> ? [V5s: tyop_2Erealax_2Ereal] :
! [V6y: tyop_2Erealax_2Ereal] :
( ? [V7x: tyop_2Erealax_2Ereal] :
( ( V0P @ V7x )
& ( c_2Erealax_2Ereal__lt @ V6y @ V7x ) )
<=> ( c_2Erealax_2Ereal__lt @ V6y @ V5s ) ) ) ).
thf(thm_2Erealax_2EREAL__10,axiom,
(~) @ ( c_2Erealax_2Ereal__1 = c_2Erealax_2Ereal__0 ) ).
thf(thm_2Erealax_2EREAL__ADD__SYM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V0x @ V1y )
= ( c_2Erealax_2Ereal__add @ V1y @ V0x ) ) ).
thf(thm_2Erealax_2EREAL__MUL__SYM,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ V1y )
= ( c_2Erealax_2Ereal__mul @ V1y @ V0x ) ) ).
thf(thm_2Erealax_2EREAL__ADD__ASSOC,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ V0x @ ( c_2Erealax_2Ereal__add @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ V2z ) ) ).
thf(thm_2Erealax_2EREAL__MUL__ASSOC,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Ereal__mul @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ V2z ) ) ).
thf(thm_2Erealax_2EREAL__LDISTRIB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ V0x @ ( c_2Erealax_2Ereal__add @ V1y @ V2z ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__mul @ V0x @ V2z ) ) ) ).
thf(thm_2Erealax_2EREAL__ADD__LID,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ c_2Erealax_2Ereal__0 @ V0x )
= V0x ) ).
thf(thm_2Erealax_2EREAL__MUL__LID,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul @ c_2Erealax_2Ereal__1 @ V0x )
= V0x ) ).
thf(thm_2Erealax_2EREAL__ADD__LINV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__neg @ V0x ) @ V0x )
= c_2Erealax_2Ereal__0 ) ).
thf(thm_2Erealax_2EREAL__MUL__LINV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( (~) @ ( V0x = c_2Erealax_2Ereal__0 ) )
=> ( ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Einv @ V0x ) @ V0x )
= c_2Erealax_2Ereal__1 ) ) ).
thf(thm_2Erealax_2EREAL__LT__TOTAL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( V0x = V1y )
| ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
| ( c_2Erealax_2Ereal__lt @ V1y @ V0x ) ) ).
thf(thm_2Erealax_2EREAL__LT__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( (~) @ ( c_2Erealax_2Ereal__lt @ V0x @ V0x ) ) ).
thf(thm_2Erealax_2EREAL__LT__TRANS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V0x @ V1y )
& ( c_2Erealax_2Ereal__lt @ V1y @ V2z ) )
=> ( c_2Erealax_2Ereal__lt @ V0x @ V2z ) ) ).
thf(thm_2Erealax_2EREAL__LT__IADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V1y @ V2z )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) @ ( c_2Erealax_2Ereal__add @ V0x @ V2z ) ) ) ).
thf(thm_2Erealax_2EREAL__LT__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ c_2Erealax_2Ereal__0 @ V0x )
& ( c_2Erealax_2Ereal__lt @ c_2Erealax_2Ereal__0 @ V1y ) )
=> ( c_2Erealax_2Ereal__lt @ c_2Erealax_2Ereal__0 @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) ) ) ).
thf(thm_2Erealax_2EREAL__INV__0,axiom,
( ( c_2Erealax_2Einv @ c_2Erealax_2Ereal__0 )
= c_2Erealax_2Ereal__0 ) ).
%------------------------------------------------------------------------------