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 : 235 ( 91 unt; 90 typ; 0 def)
% Number of atoms : 241 ( 109 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 112 ( 16 ~; 7 |; 25 &)
% ( 23 <=>; 41 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 65 ( 41 >; 24 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 86 ( 86 usr; 35 con; 0-5 aty)
% Number of variables : 334 ( 289 !; 15 ?; 334 :)
% ( 30 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Ehreal_2Ehreal,type,
tyop_2Ehreal_2Ehreal: $tType ).
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).
tff(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
tff(app_2E2,type,
app_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(p,type,
p: tyop_2Emin_2Ebool > $o ).
tff(combin_i_2E0,type,
combin_i_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(combin_k_2E0,type,
combin_k_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).
tff(combin_s_2E0,type,
combin_s_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).
tff(c_2Ebool_2E_21_2E0,type,
c_2Ebool_2E_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_21_2E1,type,
c_2Ebool_2E_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Epair_2E_2C_2E0,type,
c_2Epair_2E_2C_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))) ).
tff(c_2Epair_2E_2C_2E2,type,
c_2Epair_2E_2C_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).
tff(c_2Ebool_2E_2F_5C_2E0,type,
c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_2F_5C_2E2,type,
c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Emin_2E_3D_2E0,type,
c_2Emin_2E_3D_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_2E2,type,
c_2Emin_2E_3D_2E2:
!>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_3D_3D_3E_2E0,type,
c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_3D_3E_2E2,type,
c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2E_3F_2E0,type,
c_2Ebool_2E_3F_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_2E1,type,
c_2Ebool_2E_3F_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_40_2E0,type,
c_2Emin_2E_40_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a) ).
tff(c_2Emin_2E_40_2E1,type,
c_2Emin_2E_40_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > A_27a ) ).
tff(c_2Ebool_2ECOND_2E0,type,
c_2Ebool_2ECOND_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).
tff(c_2Ebool_2ECOND_2E3,type,
c_2Ebool_2ECOND_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Ebool * A_27a * A_27a ) > A_27a ) ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Equotient_2EQUOTIENT_2E0,type,
c_2Equotient_2EQUOTIENT_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool))) ).
tff(c_2Equotient_2EQUOTIENT_2E3,type,
c_2Equotient_2EQUOTIENT_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E0,type,
c_2Ebool_2ETYPE__DEFINITION_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
c_2Ebool_2ETYPE__DEFINITION_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_5C_2F_2E0,type,
c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_5C_2F_2E2,type,
c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ehreal_2Ehreal__1_2E0,type,
c_2Ehreal_2Ehreal__1_2E0: tyop_2Ehreal_2Ehreal ).
tff(c_2Ehreal_2Ehreal__add_2E0,type,
c_2Ehreal_2Ehreal__add_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)) ).
tff(c_2Ehreal_2Ehreal__add_2E2,type,
c_2Ehreal_2Ehreal__add_2E2: ( tyop_2Ehreal_2Ehreal * tyop_2Ehreal_2Ehreal ) > tyop_2Ehreal_2Ehreal ).
tff(c_2Ehreal_2Ehreal__inv_2E0,type,
c_2Ehreal_2Ehreal__inv_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) ).
tff(c_2Ehreal_2Ehreal__inv_2E1,type,
c_2Ehreal_2Ehreal__inv_2E1: tyop_2Ehreal_2Ehreal > tyop_2Ehreal_2Ehreal ).
tff(c_2Ehreal_2Ehreal__lt_2E0,type,
c_2Ehreal_2Ehreal__lt_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)) ).
tff(c_2Ehreal_2Ehreal__lt_2E2,type,
c_2Ehreal_2Ehreal__lt_2E2: ( tyop_2Ehreal_2Ehreal * tyop_2Ehreal_2Ehreal ) > tyop_2Emin_2Ebool ).
tff(c_2Ehreal_2Ehreal__mul_2E0,type,
c_2Ehreal_2Ehreal__mul_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)) ).
tff(c_2Ehreal_2Ehreal__mul_2E2,type,
c_2Ehreal_2Ehreal__mul_2E2: ( tyop_2Ehreal_2Ehreal * tyop_2Ehreal_2Ehreal ) > tyop_2Ehreal_2Ehreal ).
tff(c_2Erealax_2Ehreal__of__real_2E0,type,
c_2Erealax_2Ehreal__of__real_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Ehreal_2Ehreal) ).
tff(c_2Erealax_2Ehreal__of__real_2E1,type,
c_2Erealax_2Ehreal__of__real_2E1: tyop_2Erealax_2Ereal > tyop_2Ehreal_2Ehreal ).
tff(c_2Erealax_2Ehreal__of__treal_2E0,type,
c_2Erealax_2Ehreal__of__treal_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Ehreal_2Ehreal) ).
tff(c_2Erealax_2Ehreal__of__treal_2E1,type,
c_2Erealax_2Ehreal__of__treal_2E1: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) > tyop_2Ehreal_2Ehreal ).
tff(c_2Ehreal_2Ehreal__sub_2E0,type,
c_2Ehreal_2Ehreal__sub_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)) ).
tff(c_2Ehreal_2Ehreal__sub_2E2,type,
c_2Ehreal_2Ehreal__sub_2E2: ( tyop_2Ehreal_2Ehreal * tyop_2Ehreal_2Ehreal ) > tyop_2Ehreal_2Ehreal ).
tff(c_2Erealax_2Einv_2E0,type,
c_2Erealax_2Einv_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Einv_2E1,type,
c_2Erealax_2Einv_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__0_2E0,type,
c_2Erealax_2Ereal__0_2E0: tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__1_2E0,type,
c_2Erealax_2Ereal__1_2E0: tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__ABS_2E0,type,
c_2Erealax_2Ereal__ABS_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Ereal__ABS_2E1,type,
c_2Erealax_2Ereal__ABS_2E1: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__ABS__CLASS_2E0,type,
c_2Erealax_2Ereal__ABS__CLASS_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Ereal__ABS__CLASS_2E1,type,
c_2Erealax_2Ereal__ABS__CLASS_2E1: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool) > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__REP_2E0,type,
c_2Erealax_2Ereal__REP_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)) ).
tff(c_2Erealax_2Ereal__REP_2E1,type,
c_2Erealax_2Ereal__REP_2E1: tyop_2Erealax_2Ereal > tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) ).
tff(c_2Erealax_2Ereal__REP__CLASS_2E0,type,
c_2Erealax_2Ereal__REP__CLASS_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)) ).
tff(c_2Erealax_2Ereal__REP__CLASS_2E1,type,
c_2Erealax_2Ereal__REP__CLASS_2E1: tyop_2Erealax_2Ereal > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool) ).
tff(c_2Erealax_2Ereal__add_2E0,type,
c_2Erealax_2Ereal__add_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Erealax_2Ereal__add_2E2,type,
c_2Erealax_2Ereal__add_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__lt_2E0,type,
c_2Erealax_2Ereal__lt_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Erealax_2Ereal__lt_2E2,type,
c_2Erealax_2Ereal__lt_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Erealax_2Ereal__mul_2E0,type,
c_2Erealax_2Ereal__mul_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Erealax_2Ereal__mul_2E2,type,
c_2Erealax_2Ereal__mul_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__neg_2E0,type,
c_2Erealax_2Ereal__neg_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Ereal__neg_2E1,type,
c_2Erealax_2Ereal__neg_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__of__hreal_2E0,type,
c_2Erealax_2Ereal__of__hreal_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Ereal__of__hreal_2E1,type,
c_2Erealax_2Ereal__of__hreal_2E1: tyop_2Ehreal_2Ehreal > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Etreal__0_2E0,type,
c_2Erealax_2Etreal__0_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) ).
tff(c_2Erealax_2Etreal__1_2E0,type,
c_2Erealax_2Etreal__1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) ).
tff(c_2Erealax_2Etreal__add_2E0,type,
c_2Erealax_2Etreal__add_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal))) ).
tff(c_2Erealax_2Etreal__add_2E2,type,
c_2Erealax_2Etreal__add_2E2: ( 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) ).
tff(c_2Erealax_2Etreal__eq_2E0,type,
c_2Erealax_2Etreal__eq_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)) ).
tff(c_2Erealax_2Etreal__eq_2E1,type,
c_2Erealax_2Etreal__eq_2E1: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool) ).
tff(c_2Erealax_2Etreal__eq_2E2,type,
c_2Erealax_2Etreal__eq_2E2: ( tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) * tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) ) > tyop_2Emin_2Ebool ).
tff(c_2Erealax_2Etreal__inv_2E0,type,
c_2Erealax_2Etreal__inv_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)) ).
tff(c_2Erealax_2Etreal__inv_2E1,type,
c_2Erealax_2Etreal__inv_2E1: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) > tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) ).
tff(c_2Erealax_2Etreal__lt_2E0,type,
c_2Erealax_2Etreal__lt_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)) ).
tff(c_2Erealax_2Etreal__lt_2E2,type,
c_2Erealax_2Etreal__lt_2E2: ( tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) * tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) ) > tyop_2Emin_2Ebool ).
tff(c_2Erealax_2Etreal__mul_2E0,type,
c_2Erealax_2Etreal__mul_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal))) ).
tff(c_2Erealax_2Etreal__mul_2E2,type,
c_2Erealax_2Etreal__mul_2E2: ( 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) ).
tff(c_2Erealax_2Etreal__neg_2E0,type,
c_2Erealax_2Etreal__neg_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)) ).
tff(c_2Erealax_2Etreal__neg_2E1,type,
c_2Erealax_2Etreal__neg_2E1: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) > tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) ).
tff(c_2Erealax_2Etreal__of__hreal_2E0,type,
c_2Erealax_2Etreal__of__hreal_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)) ).
tff(c_2Erealax_2Etreal__of__hreal_2E1,type,
c_2Erealax_2Etreal__of__hreal_2E1: tyop_2Ehreal_2Ehreal > tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal) ).
tff(c_2Ebool_2E_7E_2E0,type,
c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_7E_2E1,type,
c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).
tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
=> ( V0f_2E0 = V1g_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Eboolext,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( ( p(V0_2E0)
<=> p(V1_2E0) )
=> ( V0_2E0 = V1_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Etruth,axiom,
p(c_2Ebool_2ET_2E0) ).
tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
~ p(c_2Ebool_2EF_2E0) ).
tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
| ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Es__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).
tff(logicdef_2E_2F_5C,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
& p(V1_2E0) ) ) ).
tff(logicdef_2E_5C_2F,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
| p(V1_2E0) ) ) ).
tff(logicdef_2E_7E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_7E_2E1(V0_2E0))
<=> ~ p(V0_2E0) ) ).
tff(logicdef_2E_3D_3D_3E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
=> p(V1_2E0) ) ) ).
tff(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
<=> ( V0_2E0 = V1_2E0 ) ) ).
tff(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
<=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
<=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Ehreal_2Ehreal_20mono_2Etyop_2Ehreal_2Ehreal,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal,X1_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,X0_2E0,X1_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),c_2Epair_2E_2C_2E0(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
& p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
( ( X0_2E0 = X1_2E0 )
<=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
=> p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Ehreal_2Ehreal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Ehreal_2Ehreal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),tyop_2Ehreal_2Ehreal,c_2Emin_2E_40_2E0(tyop_2Ehreal_2Ehreal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Ehreal_2Ehreal_2Ctyop_2Ehreal_2Ehreal_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Emin_2E_40_2E0(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),X0_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Ehreal_2Ehreal_2Ctyop_2Ehreal_2Ehreal_29,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Ebool_2ECOND_2E3(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal))),c_2Ebool_2ECOND_2E0(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Equotient_2EQUOTIENT_2E3_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Ehreal_2Ehreal_2Ctyop_2Ehreal_2Ehreal_29_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal),X2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal))] : ( c_2Equotient_2EQUOTIENT_2E3(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),tyop_2Emin_2Ebool)),c_2Equotient_2EQUOTIENT_2E0(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Epair_2Eprod_28tyop_2Ehreal_2Ehreal_2Ctyop_2Ehreal_2Ehreal_29_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),c_2Ebool_2ETYPE__DEFINITION_2E0(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
| p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Ehreal_2Ehreal__add_2E2,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal,X1_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Ehreal_2Ehreal__add_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ehreal_2Ehreal__inv_2E1,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__inv_2E1(X0_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__inv_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ehreal_2Ehreal__lt_2E2,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal,X1_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__lt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),c_2Ehreal_2Ehreal__lt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ehreal_2Ehreal__mul_2E2,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal,X1_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Ehreal_2Ehreal__mul_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Ehreal__of__real_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ehreal__of__real_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Ehreal_2Ehreal,c_2Erealax_2Ehreal__of__real_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Ehreal__of__treal_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Ehreal__of__treal_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Ehreal_2Ehreal,c_2Erealax_2Ehreal__of__treal_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ehreal_2Ehreal__sub_2E2,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal,X1_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__sub_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Ehreal_2Ehreal__sub_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Einv_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Einv_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Ereal__ABS_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Ereal__ABS_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Ereal__ABS__CLASS_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)] : ( c_2Erealax_2Ereal__ABS__CLASS_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS__CLASS_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Ereal__REP_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__REP_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Ereal__REP__CLASS_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__REP__CLASS_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Ereal__REP__CLASS_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__add_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__add_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__lt_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Erealax_2Ereal__lt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__mul_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__mul_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Ereal__neg_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Ereal__of__hreal_2E1,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Ereal__of__hreal_2E1(X0_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Etreal__add_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),c_2Erealax_2Etreal__add_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Etreal__eq_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__eq_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Etreal__eq_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Etreal__eq_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__eq_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Etreal__eq_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Etreal__inv_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__inv_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__inv_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Etreal__lt_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__lt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Etreal__lt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Etreal__mul_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),c_2Erealax_2Etreal__mul_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Etreal__neg_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__neg_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__neg_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Etreal__of__hreal_2E1,axiom,
! [X0_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Etreal__of__hreal_2E1(X0_2E0) = app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__of__hreal_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Ebool] :
( ~ p(X0_2E0)
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).
tff(thm_2Erealax_2Etreal__0,axiom,
c_2Erealax_2Etreal__0_2E0 = c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0,c_2Ehreal_2Ehreal__1_2E0) ).
tff(thm_2Erealax_2Etreal__1,axiom,
c_2Erealax_2Etreal__1_2E0 = c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(c_2Ehreal_2Ehreal__1_2E0,c_2Ehreal_2Ehreal__1_2E0),c_2Ehreal_2Ehreal__1_2E0) ).
tff(thm_2Erealax_2Etreal__neg,axiom,
! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Etreal__neg_2E1(c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,V0x_2E0,V1y_2E0)) = c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,V1y_2E0,V0x_2E0) ) ).
tff(thm_2Erealax_2Etreal__add,axiom,
! [V0x1_2E0: tyop_2Ehreal_2Ehreal,V1y1_2E0: tyop_2Ehreal_2Ehreal,V2x2_2E0: tyop_2Ehreal_2Ehreal,V3y2_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Etreal__add_2E2(c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,V0x1_2E0,V1y1_2E0),c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,V2x2_2E0,V3y2_2E0)) = c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(V0x1_2E0,V2x2_2E0),c_2Ehreal_2Ehreal__add_2E2(V1y1_2E0,V3y2_2E0)) ) ).
tff(thm_2Erealax_2Etreal__mul,axiom,
! [V0x1_2E0: tyop_2Ehreal_2Ehreal,V1y1_2E0: tyop_2Ehreal_2Ehreal,V2x2_2E0: tyop_2Ehreal_2Ehreal,V3y2_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Etreal__mul_2E2(c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,V0x1_2E0,V1y1_2E0),c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,V2x2_2E0,V3y2_2E0)) = c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(c_2Ehreal_2Ehreal__mul_2E2(V0x1_2E0,V2x2_2E0),c_2Ehreal_2Ehreal__mul_2E2(V1y1_2E0,V3y2_2E0)),c_2Ehreal_2Ehreal__add_2E2(c_2Ehreal_2Ehreal__mul_2E2(V0x1_2E0,V3y2_2E0),c_2Ehreal_2Ehreal__mul_2E2(V1y1_2E0,V2x2_2E0))) ) ).
tff(thm_2Erealax_2Etreal__lt,axiom,
! [V0x1_2E0: tyop_2Ehreal_2Ehreal,V1y1_2E0: tyop_2Ehreal_2Ehreal,V2x2_2E0: tyop_2Ehreal_2Ehreal,V3y2_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Etreal__lt_2E2(c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,V0x1_2E0,V1y1_2E0),c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,V2x2_2E0,V3y2_2E0)) = c_2Ehreal_2Ehreal__lt_2E2(c_2Ehreal_2Ehreal__add_2E2(V0x1_2E0,V3y2_2E0),c_2Ehreal_2Ehreal__add_2E2(V2x2_2E0,V1y1_2E0)) ) ).
tff(thm_2Erealax_2Etreal__inv,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal] :
( p(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1y_2E0))
<=> ( V0x_2E0 = V1y_2E0 ) )
=> ! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Etreal__inv_2E1(c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,V0x_2E0,V1y_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1y_2E0),c_2Erealax_2Etreal__0_2E0,c_2Ebool_2ECOND_2E3(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Ehreal_2Ehreal__lt_2E2(V1y_2E0,V0x_2E0),c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(c_2Ehreal_2Ehreal__inv_2E1(c_2Ehreal_2Ehreal__sub_2E2(V0x_2E0,V1y_2E0)),c_2Ehreal_2Ehreal__1_2E0),c_2Ehreal_2Ehreal__1_2E0),c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0,c_2Ehreal_2Ehreal__add_2E2(c_2Ehreal_2Ehreal__inv_2E1(c_2Ehreal_2Ehreal__sub_2E2(V1y_2E0,V0x_2E0)),c_2Ehreal_2Ehreal__1_2E0)))) ) ) ).
tff(thm_2Erealax_2Etreal__eq,axiom,
! [V0x1_2E0: tyop_2Ehreal_2Ehreal,V1y1_2E0: tyop_2Ehreal_2Ehreal,V2x2_2E0: tyop_2Ehreal_2Ehreal,V3y2_2E0: tyop_2Ehreal_2Ehreal] :
( p(c_2Erealax_2Etreal__eq_2E2(c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,V0x1_2E0,V1y1_2E0),c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,V2x2_2E0,V3y2_2E0)))
<=> ( c_2Ehreal_2Ehreal__add_2E2(V0x1_2E0,V3y2_2E0) = c_2Ehreal_2Ehreal__add_2E2(V2x2_2E0,V1y1_2E0) ) ) ).
tff(thm_2Erealax_2Etreal__of__hreal,axiom,
! [V0x_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Etreal__of__hreal_2E1(V0x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(V0x_2E0,c_2Ehreal_2Ehreal__1_2E0),c_2Ehreal_2Ehreal__1_2E0) ) ).
tff(thm_2Erealax_2Ehreal__of__treal,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)))] :
( ! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal,V2d_2E0: tyop_2Ehreal_2Ehreal] :
( p(app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)),F0_2E0,V0x_2E0),V1y_2E0),V2d_2E0))
<=> ( V0x_2E0 = c_2Ehreal_2Ehreal__add_2E2(V1y_2E0,V2d_2E0) ) )
=> ! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Ehreal__of__treal_2E1(c_2Epair_2E_2C_2E2(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal,V0x_2E0,V1y_2E0)) = c_2Emin_2E_40_2E1(tyop_2Ehreal_2Ehreal,app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),app_2E2(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)),F0_2E0,V0x_2E0),V1y_2E0)) ) ) ).
tff(thm_2Erealax_2Ereal__TY__DEF,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V1c_2E0))
<=> ? [V2r_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__eq_2E2(V2r_2E0,V2r_2E0))
& ( V1c_2E0 = c_2Erealax_2Etreal__eq_2E1(V2r_2E0) ) ) )
=> ? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,F0_2E0,V0rep_2E0)) ) ).
tff(thm_2Erealax_2Ereal__bijections,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__ABS__CLASS_2E1(c_2Erealax_2Ereal__REP__CLASS_2E1(V0a_2E0)) = V0a_2E0 )
& ! [V1r_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)] :
( ? [V2r_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__eq_2E2(V2r_2E0,V2r_2E0))
& ( V1r_2E0 = c_2Erealax_2Etreal__eq_2E1(V2r_2E0) ) )
<=> ( c_2Erealax_2Ereal__REP__CLASS_2E1(c_2Erealax_2Ereal__ABS__CLASS_2E1(V1r_2E0)) = V1r_2E0 ) ) ) ).
tff(thm_2Erealax_2Ereal__REP__def,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__REP_2E1(V0a_2E0) = c_2Emin_2E_40_2E1(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP__CLASS_2E1(V0a_2E0)) ) ).
tff(thm_2Erealax_2Ereal__ABS__def,axiom,
! [V0r_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Ereal__ABS_2E1(V0r_2E0) = c_2Erealax_2Ereal__ABS__CLASS_2E1(c_2Erealax_2Etreal__eq_2E1(V0r_2E0)) ) ).
tff(thm_2Erealax_2Ereal__0,axiom,
c_2Erealax_2Ereal__0_2E0 = c_2Erealax_2Ereal__ABS_2E1(c_2Erealax_2Etreal__0_2E0) ).
tff(thm_2Erealax_2Ereal__1,axiom,
c_2Erealax_2Ereal__1_2E0 = c_2Erealax_2Ereal__ABS_2E1(c_2Erealax_2Etreal__1_2E0) ).
tff(thm_2Erealax_2Ereal__neg,axiom,
! [V0T1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(V0T1_2E0) = c_2Erealax_2Ereal__ABS_2E1(c_2Erealax_2Etreal__neg_2E1(c_2Erealax_2Ereal__REP_2E1(V0T1_2E0))) ) ).
tff(thm_2Erealax_2Ereal__inv,axiom,
! [V0T1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Einv_2E1(V0T1_2E0) = c_2Erealax_2Ereal__ABS_2E1(c_2Erealax_2Etreal__inv_2E1(c_2Erealax_2Ereal__REP_2E1(V0T1_2E0))) ) ).
tff(thm_2Erealax_2Ereal__add,axiom,
! [V0T1_2E0: tyop_2Erealax_2Ereal,V1T2_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(V0T1_2E0,V1T2_2E0) = c_2Erealax_2Ereal__ABS_2E1(c_2Erealax_2Etreal__add_2E2(c_2Erealax_2Ereal__REP_2E1(V0T1_2E0),c_2Erealax_2Ereal__REP_2E1(V1T2_2E0))) ) ).
tff(thm_2Erealax_2Ereal__mul,axiom,
! [V0T1_2E0: tyop_2Erealax_2Ereal,V1T2_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(V0T1_2E0,V1T2_2E0) = c_2Erealax_2Ereal__ABS_2E1(c_2Erealax_2Etreal__mul_2E2(c_2Erealax_2Ereal__REP_2E1(V0T1_2E0),c_2Erealax_2Ereal__REP_2E1(V1T2_2E0))) ) ).
tff(thm_2Erealax_2Ereal__lt,axiom,
! [V0T1_2E0: tyop_2Erealax_2Ereal,V1T2_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(V0T1_2E0,V1T2_2E0) = c_2Erealax_2Etreal__lt_2E2(c_2Erealax_2Ereal__REP_2E1(V0T1_2E0),c_2Erealax_2Ereal__REP_2E1(V1T2_2E0)) ) ).
tff(thm_2Erealax_2Ereal__of__hreal,axiom,
! [V0T1_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Ereal__of__hreal_2E1(V0T1_2E0) = c_2Erealax_2Ereal__ABS_2E1(c_2Erealax_2Etreal__of__hreal_2E1(V0T1_2E0)) ) ).
tff(thm_2Erealax_2Ehreal__of__real,axiom,
! [V0T1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ehreal__of__real_2E1(V0T1_2E0) = c_2Erealax_2Ehreal__of__treal_2E1(c_2Erealax_2Ereal__REP_2E1(V0T1_2E0)) ) ).
tff(thm_2Erealax_2EHREAL__RDISTRIB,axiom,
! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal,V2z_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__mul_2E2(c_2Ehreal_2Ehreal__add_2E2(V0x_2E0,V1y_2E0),V2z_2E0) = c_2Ehreal_2Ehreal__add_2E2(c_2Ehreal_2Ehreal__mul_2E2(V0x_2E0,V2z_2E0),c_2Ehreal_2Ehreal__mul_2E2(V1y_2E0,V2z_2E0)) ) ).
tff(thm_2Erealax_2EHREAL__EQ__ADDR,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal,V1Y_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__add_2E2(V0X_2E0,V1Y_2E0) != V0X_2E0 ) ).
tff(thm_2Erealax_2EHREAL__EQ__ADDL,axiom,
! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal] : ( V0x_2E0 != c_2Ehreal_2Ehreal__add_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Erealax_2EHREAL__EQ__LADD,axiom,
! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal,V2z_2E0: tyop_2Ehreal_2Ehreal] :
( ( c_2Ehreal_2Ehreal__add_2E2(V0x_2E0,V1y_2E0) = c_2Ehreal_2Ehreal__add_2E2(V0x_2E0,V2z_2E0) )
<=> ( V1y_2E0 = V2z_2E0 ) ) ).
tff(thm_2Erealax_2EHREAL__LT__REFL,axiom,
! [V0x_2E0: tyop_2Ehreal_2Ehreal] : ~ p(c_2Ehreal_2Ehreal__lt_2E2(V0x_2E0,V0x_2E0)) ).
tff(thm_2Erealax_2EHREAL__LT__ADDL,axiom,
! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal] : p(c_2Ehreal_2Ehreal__lt_2E2(V0x_2E0,c_2Ehreal_2Ehreal__add_2E2(V0x_2E0,V1y_2E0))) ).
tff(thm_2Erealax_2EHREAL__LT__NE,axiom,
! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal] :
( p(c_2Ehreal_2Ehreal__lt_2E2(V0x_2E0,V1y_2E0))
=> ( V0x_2E0 != V1y_2E0 ) ) ).
tff(thm_2Erealax_2EHREAL__LT__ADDR,axiom,
! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal] : ~ p(c_2Ehreal_2Ehreal__lt_2E2(c_2Ehreal_2Ehreal__add_2E2(V0x_2E0,V1y_2E0),V0x_2E0)) ).
tff(thm_2Erealax_2EHREAL__LT__GT,axiom,
! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal] :
( p(c_2Ehreal_2Ehreal__lt_2E2(V0x_2E0,V1y_2E0))
=> ~ p(c_2Ehreal_2Ehreal__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Erealax_2EHREAL__LT__ADD2,axiom,
! [V0x1_2E0: tyop_2Ehreal_2Ehreal,V1x2_2E0: tyop_2Ehreal_2Ehreal,V2y1_2E0: tyop_2Ehreal_2Ehreal,V3y2_2E0: tyop_2Ehreal_2Ehreal] :
( ( p(c_2Ehreal_2Ehreal__lt_2E2(V0x1_2E0,V2y1_2E0))
& p(c_2Ehreal_2Ehreal__lt_2E2(V1x2_2E0,V3y2_2E0)) )
=> p(c_2Ehreal_2Ehreal__lt_2E2(c_2Ehreal_2Ehreal__add_2E2(V0x1_2E0,V1x2_2E0),c_2Ehreal_2Ehreal__add_2E2(V2y1_2E0,V3y2_2E0))) ) ).
tff(thm_2Erealax_2EHREAL__LT__LADD,axiom,
! [V0x_2E0: tyop_2Ehreal_2Ehreal,V1y_2E0: tyop_2Ehreal_2Ehreal,V2z_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__lt_2E2(c_2Ehreal_2Ehreal__add_2E2(V0x_2E0,V1y_2E0),c_2Ehreal_2Ehreal__add_2E2(V0x_2E0,V2z_2E0)) = c_2Ehreal_2Ehreal__lt_2E2(V1y_2E0,V2z_2E0) ) ).
tff(thm_2Erealax_2ETREAL__EQ__REFL,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : p(c_2Erealax_2Etreal__eq_2E2(V0x_2E0,V0x_2E0)) ).
tff(thm_2Erealax_2ETREAL__EQ__SYM,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__eq_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Etreal__eq_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Erealax_2ETREAL__EQ__TRANS,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( ( p(c_2Erealax_2Etreal__eq_2E2(V0x_2E0,V1y_2E0))
& p(c_2Erealax_2Etreal__eq_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Erealax_2Etreal__eq_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Erealax_2ETREAL__EQ__EQUIV,axiom,
! [V0p_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1q_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__eq_2E2(V0p_2E0,V1q_2E0))
<=> ( c_2Erealax_2Etreal__eq_2E1(V0p_2E0) = c_2Erealax_2Etreal__eq_2E1(V1q_2E0) ) ) ).
tff(thm_2Erealax_2ETREAL__EQ__AP,axiom,
! [V0p_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1q_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( ( V0p_2E0 = V1q_2E0 )
=> p(c_2Erealax_2Etreal__eq_2E2(V0p_2E0,V1q_2E0)) ) ).
tff(thm_2Erealax_2ETREAL__10,axiom,
~ p(c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__1_2E0,c_2Erealax_2Etreal__0_2E0)) ).
tff(thm_2Erealax_2ETREAL__ADD__SYM,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__add_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Etreal__add_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Erealax_2ETREAL__MUL__SYM,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Etreal__mul_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Erealax_2ETREAL__ADD__ASSOC,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__add_2E2(V0x_2E0,c_2Erealax_2Etreal__add_2E2(V1y_2E0,V2z_2E0)) = c_2Erealax_2Etreal__add_2E2(c_2Erealax_2Etreal__add_2E2(V0x_2E0,V1y_2E0),V2z_2E0) ) ).
tff(thm_2Erealax_2ETREAL__MUL__ASSOC,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__mul_2E2(V0x_2E0,c_2Erealax_2Etreal__mul_2E2(V1y_2E0,V2z_2E0)) = c_2Erealax_2Etreal__mul_2E2(c_2Erealax_2Etreal__mul_2E2(V0x_2E0,V1y_2E0),V2z_2E0) ) ).
tff(thm_2Erealax_2ETREAL__LDISTRIB,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__mul_2E2(V0x_2E0,c_2Erealax_2Etreal__add_2E2(V1y_2E0,V2z_2E0)) = c_2Erealax_2Etreal__add_2E2(c_2Erealax_2Etreal__mul_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Etreal__mul_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Erealax_2ETREAL__ADD__LID,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : p(c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__add_2E2(c_2Erealax_2Etreal__0_2E0,V0x_2E0),V0x_2E0)) ).
tff(thm_2Erealax_2ETREAL__MUL__LID,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : p(c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__mul_2E2(c_2Erealax_2Etreal__1_2E0,V0x_2E0),V0x_2E0)) ).
tff(thm_2Erealax_2ETREAL__ADD__LINV,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : p(c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__add_2E2(c_2Erealax_2Etreal__neg_2E1(V0x_2E0),V0x_2E0),c_2Erealax_2Etreal__0_2E0)) ).
tff(thm_2Erealax_2ETREAL__INV__0,axiom,
p(c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__inv_2E1(c_2Erealax_2Etreal__0_2E0),c_2Erealax_2Etreal__0_2E0)) ).
tff(thm_2Erealax_2ETREAL__MUL__LINV,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( ~ p(c_2Erealax_2Etreal__eq_2E2(V0x_2E0,c_2Erealax_2Etreal__0_2E0))
=> p(c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__mul_2E2(c_2Erealax_2Etreal__inv_2E1(V0x_2E0),V0x_2E0),c_2Erealax_2Etreal__1_2E0)) ) ).
tff(thm_2Erealax_2ETREAL__LT__TOTAL,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__eq_2E2(V0x_2E0,V1y_2E0))
| p(c_2Erealax_2Etreal__lt_2E2(V0x_2E0,V1y_2E0))
| p(c_2Erealax_2Etreal__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Erealax_2ETREAL__LT__REFL,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ~ p(c_2Erealax_2Etreal__lt_2E2(V0x_2E0,V0x_2E0)) ).
tff(thm_2Erealax_2ETREAL__LT__TRANS,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( ( p(c_2Erealax_2Etreal__lt_2E2(V0x_2E0,V1y_2E0))
& p(c_2Erealax_2Etreal__lt_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Erealax_2Etreal__lt_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Erealax_2ETREAL__LT__ADD,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__lt_2E2(V1y_2E0,V2z_2E0))
=> p(c_2Erealax_2Etreal__lt_2E2(c_2Erealax_2Etreal__add_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Etreal__add_2E2(V0x_2E0,V2z_2E0))) ) ).
tff(thm_2Erealax_2ETREAL__LT__MUL,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( ( p(c_2Erealax_2Etreal__lt_2E2(c_2Erealax_2Etreal__0_2E0,V0x_2E0))
& p(c_2Erealax_2Etreal__lt_2E2(c_2Erealax_2Etreal__0_2E0,V1y_2E0)) )
=> p(c_2Erealax_2Etreal__lt_2E2(c_2Erealax_2Etreal__0_2E0,c_2Erealax_2Etreal__mul_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Erealax_2ETREAL__BIJ,axiom,
( ! [V0h_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Erealax_2Ehreal__of__treal_2E1(c_2Erealax_2Etreal__of__hreal_2E1(V0h_2E0)) = V0h_2E0 )
& ! [V1r_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] : ( c_2Erealax_2Etreal__lt_2E2(c_2Erealax_2Etreal__0_2E0,V1r_2E0) = c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__of__hreal_2E1(c_2Erealax_2Ehreal__of__treal_2E1(V1r_2E0)),V1r_2E0) ) ) ).
tff(thm_2Erealax_2ETREAL__ISO,axiom,
! [V0h_2E0: tyop_2Ehreal_2Ehreal,V1i_2E0: tyop_2Ehreal_2Ehreal] :
( p(c_2Ehreal_2Ehreal__lt_2E2(V0h_2E0,V1i_2E0))
=> p(c_2Erealax_2Etreal__lt_2E2(c_2Erealax_2Etreal__of__hreal_2E1(V0h_2E0),c_2Erealax_2Etreal__of__hreal_2E1(V1i_2E0))) ) ).
tff(thm_2Erealax_2ETREAL__BIJ__WELLDEF,axiom,
! [V0h_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1i_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__eq_2E2(V0h_2E0,V1i_2E0))
=> ( c_2Erealax_2Ehreal__of__treal_2E1(V0h_2E0) = c_2Erealax_2Ehreal__of__treal_2E1(V1i_2E0) ) ) ).
tff(thm_2Erealax_2ETREAL__NEG__WELLDEF,axiom,
! [V0x1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__eq_2E2(V0x1_2E0,V1x2_2E0))
=> p(c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__neg_2E1(V0x1_2E0),c_2Erealax_2Etreal__neg_2E1(V1x2_2E0))) ) ).
tff(thm_2Erealax_2ETREAL__ADD__WELLDEFR,axiom,
! [V0x1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__eq_2E2(V0x1_2E0,V1x2_2E0))
=> p(c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__add_2E2(V0x1_2E0,V2y_2E0),c_2Erealax_2Etreal__add_2E2(V1x2_2E0,V2y_2E0))) ) ).
tff(thm_2Erealax_2ETREAL__ADD__WELLDEF,axiom,
! [V0x1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V3y2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( ( p(c_2Erealax_2Etreal__eq_2E2(V0x1_2E0,V1x2_2E0))
& p(c_2Erealax_2Etreal__eq_2E2(V2y1_2E0,V3y2_2E0)) )
=> p(c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__add_2E2(V0x1_2E0,V2y1_2E0),c_2Erealax_2Etreal__add_2E2(V1x2_2E0,V3y2_2E0))) ) ).
tff(thm_2Erealax_2ETREAL__MUL__WELLDEFR,axiom,
! [V0x1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__eq_2E2(V0x1_2E0,V1x2_2E0))
=> p(c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__mul_2E2(V0x1_2E0,V2y_2E0),c_2Erealax_2Etreal__mul_2E2(V1x2_2E0,V2y_2E0))) ) ).
tff(thm_2Erealax_2ETREAL__MUL__WELLDEF,axiom,
! [V0x1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V3y2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( ( p(c_2Erealax_2Etreal__eq_2E2(V0x1_2E0,V1x2_2E0))
& p(c_2Erealax_2Etreal__eq_2E2(V2y1_2E0,V3y2_2E0)) )
=> p(c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__mul_2E2(V0x1_2E0,V2y1_2E0),c_2Erealax_2Etreal__mul_2E2(V1x2_2E0,V3y2_2E0))) ) ).
tff(thm_2Erealax_2ETREAL__LT__WELLDEFR,axiom,
! [V0x1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__eq_2E2(V0x1_2E0,V1x2_2E0))
=> ( c_2Erealax_2Etreal__lt_2E2(V0x1_2E0,V2y_2E0) = c_2Erealax_2Etreal__lt_2E2(V1x2_2E0,V2y_2E0) ) ) ).
tff(thm_2Erealax_2ETREAL__LT__WELLDEFL,axiom,
! [V0x_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__eq_2E2(V1y1_2E0,V2y2_2E0))
=> ( c_2Erealax_2Etreal__lt_2E2(V0x_2E0,V1y1_2E0) = c_2Erealax_2Etreal__lt_2E2(V0x_2E0,V2y2_2E0) ) ) ).
tff(thm_2Erealax_2ETREAL__LT__WELLDEF,axiom,
! [V0x1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V3y2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( ( p(c_2Erealax_2Etreal__eq_2E2(V0x1_2E0,V1x2_2E0))
& p(c_2Erealax_2Etreal__eq_2E2(V2y1_2E0,V3y2_2E0)) )
=> ( c_2Erealax_2Etreal__lt_2E2(V0x1_2E0,V2y1_2E0) = c_2Erealax_2Etreal__lt_2E2(V1x2_2E0,V3y2_2E0) ) ) ).
tff(thm_2Erealax_2ETREAL__INV__WELLDEF,axiom,
! [V0x1_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__eq_2E2(V0x1_2E0,V1x2_2E0))
=> p(c_2Erealax_2Etreal__eq_2E2(c_2Erealax_2Etreal__inv_2E1(V0x1_2E0),c_2Erealax_2Etreal__inv_2E1(V1x2_2E0))) ) ).
tff(thm_2Erealax_2Ereal__ABS__REP__CLASS,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__ABS__CLASS_2E1(c_2Erealax_2Ereal__REP__CLASS_2E1(V0a_2E0)) = V0a_2E0 )
& ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)] :
( ? [V2r_2E0: tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)] :
( p(c_2Erealax_2Etreal__eq_2E2(V2r_2E0,V2r_2E0))
& ( V1c_2E0 = c_2Erealax_2Etreal__eq_2E1(V2r_2E0) ) )
<=> ( c_2Erealax_2Ereal__REP__CLASS_2E1(c_2Erealax_2Ereal__ABS__CLASS_2E1(V1c_2E0)) = V1c_2E0 ) ) ) ).
tff(thm_2Erealax_2Ereal__QUOTIENT,axiom,
p(c_2Equotient_2EQUOTIENT_2E3(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal,c_2Erealax_2Etreal__eq_2E0,c_2Erealax_2Ereal__ABS_2E0,c_2Erealax_2Ereal__REP_2E0)) ).
tff(thm_2Erealax_2EREAL__ISO__EQ,axiom,
! [V0h_2E0: tyop_2Ehreal_2Ehreal,V1i_2E0: tyop_2Ehreal_2Ehreal] : ( c_2Ehreal_2Ehreal__lt_2E2(V0h_2E0,V1i_2E0) = c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__of__hreal_2E1(V0h_2E0),c_2Erealax_2Ereal__of__hreal_2E1(V1i_2E0)) ) ).
tff(thm_2Erealax_2EREAL__POS,axiom,
! [V0X_2E0: tyop_2Ehreal_2Ehreal] : p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__0_2E0,c_2Erealax_2Ereal__of__hreal_2E1(V0X_2E0))) ).
tff(thm_2Erealax_2ESUP__ALLPOS__LEMMA1,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1y_2E0: tyop_2Erealax_2Ereal] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V2x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__0_2E0,V2x_2E0)) )
=> ( ? [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V3x_2E0)) )
<=> ? [V4X_2E0: tyop_2Ehreal_2Ehreal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Erealax_2Ereal__of__hreal_2E1(V4X_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,c_2Erealax_2Ereal__of__hreal_2E1(V4X_2E0))) ) ) ) ).
tff(thm_2Erealax_2ESUP__ALLPOS__LEMMA2,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1X_2E0: tyop_2Ehreal_2Ehreal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Erealax_2Ereal__of__hreal_2E1(V1X_2E0)) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Erealax_2Ereal__of__hreal_2E1(V1X_2E0)) ) ).
tff(thm_2Erealax_2ESUP__ALLPOS__LEMMA3,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V1x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__0_2E0,V1x_2E0)) )
& ? [V2x_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V2x_2E0))
& ? [V3z_2E0: tyop_2Erealax_2Ereal] :
! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V4x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(V4x_2E0,V3z_2E0)) ) )
=> ( ? [V5X_2E0: tyop_2Ehreal_2Ehreal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Erealax_2Ereal__of__hreal_2E1(V5X_2E0)))
& ? [V6Y_2E0: tyop_2Ehreal_2Ehreal] :
! [V7X_2E0: tyop_2Ehreal_2Ehreal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Erealax_2Ereal__of__hreal_2E1(V7X_2E0)))
=> p(c_2Ehreal_2Ehreal__lt_2E2(V7X_2E0,V6Y_2E0)) ) ) ) ).
tff(thm_2Erealax_2ESUP__ALLPOS__LEMMA4,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( ~ p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__0_2E0,V0y_2E0))
=> ! [V1x_2E0: tyop_2Ehreal_2Ehreal] : p(c_2Erealax_2Ereal__lt_2E2(V0y_2E0,c_2Erealax_2Ereal__of__hreal_2E1(V1x_2E0))) ) ).
tff(thm_2Erealax_2EREAL__SUP__ALLPOS,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V1x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__0_2E0,V1x_2E0)) )
& ? [V2x_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V2x_2E0))
& ? [V3z_2E0: tyop_2Erealax_2Ereal] :
! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V4x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(V4x_2E0,V3z_2E0)) ) )
=> ? [V5s_2E0: tyop_2Erealax_2Ereal] :
! [V6y_2E0: tyop_2Erealax_2Ereal] :
( ? [V7x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V7x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V6y_2E0,V7x_2E0)) )
<=> p(c_2Erealax_2Ereal__lt_2E2(V6y_2E0,V5s_2E0)) ) ) ).
tff(thm_2Erealax_2EREAL__10,axiom,
c_2Erealax_2Ereal__1_2E0 != c_2Erealax_2Ereal__0_2E0 ).
tff(thm_2Erealax_2EREAL__ADD__SYM,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__add_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Erealax_2EREAL__MUL__SYM,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Erealax_2EREAL__ADD__ASSOC,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(V0x_2E0,c_2Erealax_2Ereal__add_2E2(V1y_2E0,V2z_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),V2z_2E0) ) ).
tff(thm_2Erealax_2EREAL__MUL__ASSOC,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),V2z_2E0) ) ).
tff(thm_2Erealax_2EREAL__LDISTRIB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,c_2Erealax_2Ereal__add_2E2(V1y_2E0,V2z_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Erealax_2EREAL__ADD__LID,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__0_2E0,V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Erealax_2EREAL__MUL__LID,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__1_2E0,V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Erealax_2EREAL__ADD__LINV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),V0x_2E0) = c_2Erealax_2Ereal__0_2E0 ) ).
tff(thm_2Erealax_2EREAL__MUL__LINV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Erealax_2Ereal__0_2E0 )
=> ( c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(V0x_2E0),V0x_2E0) = c_2Erealax_2Ereal__1_2E0 ) ) ).
tff(thm_2Erealax_2EREAL__LT__TOTAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 = V1y_2E0 )
| p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
| p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Erealax_2EREAL__LT__REFL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V0x_2E0)) ).
tff(thm_2Erealax_2EREAL__LT__TRANS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Erealax_2EREAL__LT__IADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V2z_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2z_2E0))) ) ).
tff(thm_2Erealax_2EREAL__LT__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__0_2E0,V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__0_2E0,V1y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__0_2E0,c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Erealax_2EREAL__INV__0,axiom,
c_2Erealax_2Einv_2E1(c_2Erealax_2Ereal__0_2E0) = c_2Erealax_2Ereal__0_2E0 ).
%------------------------------------------------------------------------------