ITP001 Axioms: ITP118_7.ax
%------------------------------------------------------------------------------
% File : ITP118_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 : real.ax [Gau19]
% : HL4118_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 645 ( 245 unt; 117 typ; 0 def)
% Number of atoms : 1105 ( 544 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 637 ( 60 ~; 25 |; 176 &)
% ( 117 <=>; 259 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 4 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 85 ( 54 >; 31 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 113 ( 113 usr; 46 con; 0-4 aty)
% Number of variables : 1398 (1303 !; 62 ?;1398 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $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_2Earithmetic_2E_2A_2E0,type,
c_2Earithmetic_2E_2A_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2E_2A_2E2,type,
c_2Earithmetic_2E_2A_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2E_2B_2E0,type,
c_2Earithmetic_2E_2B_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2E_2B_2E2,type,
c_2Earithmetic_2E_2B_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
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_2Earithmetic_2E_2D_2E0,type,
c_2Earithmetic_2E_2D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2E_2D_2E2,type,
c_2Earithmetic_2E_2D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Ereal_2E_2F_2E0,type,
c_2Ereal_2E_2F_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2E_2F_2E2,type,
c_2Ereal_2E_2F_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
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_2Enum_2E0_2E0,type,
c_2Enum_2E0_2E0: tyop_2Enum_2Enum ).
tff(c_2Eprim__rec_2E_3C_2E0,type,
c_2Eprim__rec_2E_3C_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Eprim__rec_2E_3C_2E2,type,
c_2Eprim__rec_2E_3C_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2E_3C_3D_2E0,type,
c_2Earithmetic_2E_3C_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Earithmetic_2E_3C_3D_2E2,type,
c_2Earithmetic_2E_3C_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > 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_2Earithmetic_2E_3E_3D_2E0,type,
c_2Earithmetic_2E_3E_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Earithmetic_2E_3E_3D_2E2,type,
c_2Earithmetic_2E_3E_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > 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_2Ebool_2E_3F_21_2E0,type,
c_2Ebool_2E_3F_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_21_2E1,type,
c_2Ebool_2E_3F_21_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_2Earithmetic_2EBIT1_2E0,type,
c_2Earithmetic_2EBIT1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EBIT1_2E1,type,
c_2Earithmetic_2EBIT1_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EBIT2_2E0,type,
c_2Earithmetic_2EBIT2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EBIT2_2E1,type,
c_2Earithmetic_2EBIT2_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_2Earithmetic_2EDIV_2E0,type,
c_2Earithmetic_2EDIV_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2EDIV_2E2,type,
c_2Earithmetic_2EDIV_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EEXP_2E0,type,
c_2Earithmetic_2EEXP_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2EEXP_2E2,type,
c_2Earithmetic_2EEXP_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2EFACT_2E0,type,
c_2Earithmetic_2EFACT_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EFACT_2E1,type,
c_2Earithmetic_2EFACT_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Epred__set_2EFINITE_2E0,type,
c_2Epred__set_2EFINITE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Epred__set_2EFINITE_2E1,type,
c_2Epred__set_2EFINITE_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ewhile_2ELEAST_2E0,type,
c_2Ewhile_2ELEAST_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum) ).
tff(c_2Ewhile_2ELEAST_2E1,type,
c_2Ewhile_2ELEAST_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) > tyop_2Enum_2Enum ).
tff(c_2Ebool_2ELET_2E0,type,
c_2Ebool_2ELET_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)) ).
tff(c_2Ebool_2ELET_2E2,type,
c_2Ebool_2ELET_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(c_2Earithmetic_2ENUMERAL_2E0,type,
c_2Earithmetic_2ENUMERAL_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2ENUMERAL_2E1,type,
c_2Earithmetic_2ENUMERAL_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ereal_2ENUM__CEILING_2E0,type,
c_2Ereal_2ENUM__CEILING_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum) ).
tff(c_2Ereal_2ENUM__CEILING_2E1,type,
c_2Ereal_2ENUM__CEILING_2E1: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum ).
tff(c_2Ereal_2ENUM__FLOOR_2E0,type,
c_2Ereal_2ENUM__FLOOR_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum) ).
tff(c_2Ereal_2ENUM__FLOOR_2E1,type,
c_2Ereal_2ENUM__FLOOR_2E1: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EODD_2E0,type,
c_2Earithmetic_2EODD_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).
tff(c_2Earithmetic_2EODD_2E1,type,
c_2Earithmetic_2EODD_2E1: tyop_2Enum_2Enum > tyop_2Emin_2Ebool ).
tff(c_2Enum_2ESUC_2E0,type,
c_2Enum_2ESUC_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Enum_2ESUC_2E1,type,
c_2Enum_2ESUC_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Epred__set_2EUNIV_2E0,type,
c_2Epred__set_2EUNIV_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).
tff(c_2Earithmetic_2EZERO_2E0,type,
c_2Earithmetic_2EZERO_2E0: tyop_2Enum_2Enum ).
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_2Ereal_2Eabs_2E0,type,
c_2Ereal_2Eabs_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Eabs_2E1,type,
c_2Ereal_2Eabs_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Einf_2E0,type,
c_2Ereal_2Einf_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Einf_2E1,type,
c_2Ereal_2Einf_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Erealax_2Ereal ).
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_2Ereal_2Emax_2E0,type,
c_2Ereal_2Emax_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Emax_2E2,type,
c_2Ereal_2Emax_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Emin_2E0,type,
c_2Ereal_2Emin_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Emin_2E2,type,
c_2Ereal_2Emin_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Epos_2E0,type,
c_2Ereal_2Epos_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Epos_2E1,type,
c_2Ereal_2Epos_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Epow_2E0,type,
c_2Ereal_2Epow_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Epow_2E2,type,
c_2Ereal_2Epow_2E2: ( tyop_2Erealax_2Ereal * tyop_2Enum_2Enum ) > 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__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_2Ereal_2Ereal__ge_2E0,type,
c_2Ereal_2Ereal__ge_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal_2Ereal__ge_2E2,type,
c_2Ereal_2Ereal__ge_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Ereal__gt_2E0,type,
c_2Ereal_2Ereal__gt_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal_2Ereal__gt_2E2,type,
c_2Ereal_2Ereal__gt_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
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_2Ereal_2Ereal__lte_2E0,type,
c_2Ereal_2Ereal__lte_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal_2Ereal__lte_2E2,type,
c_2Ereal_2Ereal__lte_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_2Ereal_2Ereal__of__num_2E0,type,
c_2Ereal_2Ereal__of__num_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Ereal__of__num_2E1,type,
c_2Ereal_2Ereal__of__num_2E1: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Ereal__sub_2E0,type,
c_2Ereal_2Ereal__sub_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Ereal__sub_2E2,type,
c_2Ereal_2Ereal__sub_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Esum_2E0,type,
c_2Ereal_2Esum_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Esum_2E2,type,
c_2Ereal_2Esum_2E2: ( tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) ) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Esup_2E0,type,
c_2Ereal_2Esup_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Esup_2E1,type,
c_2Ereal_2Esup_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Erealax_2Ereal ).
tff(c_2Emarker_2Eunint_2E0,type,
c_2Emarker_2Eunint_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(c_2Emarker_2Eunint_2E1,type,
c_2Emarker_2Eunint_2E1:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
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_2Earithmetic_2E_2A_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2A_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_2B_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2B_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_2D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2D_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2E_2F_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_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_2Ereal_2E_2F_2E0,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_2Eprim__rec_2E_3C_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Eprim__rec_2E_3C_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_3C_3D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3C_3D_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(arityeq2_2Ec_2Earithmetic_2E_3E_3D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3E_3D_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_2Ebool_2E_3F_21_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_21_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Emin_2E_40_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EBIT1_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT1_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EBIT2_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT2_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E0,X0_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Emin_2Ebool,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool,X2_2E0: tyop_2Emin_2Ebool] : ( c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),c_2Ebool_2ECOND_2E0(tyop_2Emin_2Ebool),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Erealax_2Ereal_29,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),c_2Ebool_2ECOND_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Ebool_2ECOND_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ebool_2ECOND_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2EDIV_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EDIV_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EDIV_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2EEXP_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEXP_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EEXP_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EFACT_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EFACT_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EFACT_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ewhile_2ELEAST_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ewhile_2ELEAST_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Ewhile_2ELEAST_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ELET_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0: tyop_2Enum_2Enum] : ( c_2Ebool_2ELET_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Ebool_2ELET_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2ENUMERAL_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENUMERAL_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal_2ENUM__CEILING_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2ENUM__CEILING_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,c_2Ereal_2ENUM__CEILING_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal_2ENUM__FLOOR_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2ENUM__FLOOR_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,c_2Ereal_2ENUM__FLOOR_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EODD_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EODD_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,c_2Earithmetic_2EODD_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,X0_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(arityeq1_2Ec_2Ereal_2Eabs_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Eabs_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal_2Einf_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal_2Einf_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Ereal_2Einf_2E0,X0_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(arityeq2_2Ec_2Ereal_2Emax_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emax_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_2Ereal_2Emax_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Emin_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emin_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_2Ereal_2Emin_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal_2Epos_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Epos_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Epos_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Epow_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),c_2Ereal_2Epow_2E0,X0_2E0),X1_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_2Ereal_2Ereal__ge_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__ge_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_2Ereal_2Ereal__ge_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__gt_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__gt_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_2Ereal_2Ereal__gt_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_2Ereal_2Ereal__lte_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_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_2Ereal_2Ereal__lte_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_2Ereal_2Ereal__of__num_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__sub_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_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_2Ereal_2Ereal__sub_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Esum_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal_2Esum_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Ereal_2Esum_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal_2Esup_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal_2Esup_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Ereal_2Esup_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Emarker_2Eunint_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Emarker_2Eunint_2E0(tyop_2Erealax_2Ereal),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_2Ereal_2Ereal__of__num,axiom,
( ( c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) = c_2Erealax_2Ereal__0_2E0 )
& ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Erealax_2Ereal__1_2E0) ) ) ).
tff(thm_2Ereal_2Ereal__sub,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__add_2E2(V0x_2E0,c_2Erealax_2Ereal__neg_2E1(V1y_2E0)) ) ).
tff(thm_2Ereal_2Ereal__lte,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
<=> ~ p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal_2Ereal__gt,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__gt_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Ereal_2Ereal__ge,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__ge_2E2(V0x_2E0,V1y_2E0) = c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Ereal_2Ereal__div,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__mul_2E2(V0x_2E0,c_2Erealax_2Einv_2E1(V1y_2E0)) ) ).
tff(thm_2Ereal_2Eabs,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(V0x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0),V0x_2E0,c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) ) ).
tff(thm_2Ereal_2Epow,axiom,
( ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Epow_2E2(V0x_2E0,c_2Enum_2E0_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
& ! [V1x_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(V1x_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)) = c_2Erealax_2Ereal__mul_2E2(V1x_2E0,c_2Ereal_2Epow_2E2(V1x_2E0,V2n_2E0)) ) ) ).
tff(thm_2Ereal_2Esup,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0P_2E0),V1s_2E0))
<=> ! [V2y_2E0: tyop_2Erealax_2Ereal] :
( ? [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2y_2E0,V3x_2E0)) )
<=> p(c_2Erealax_2Ereal__lt_2E2(V2y_2E0,V1s_2E0)) ) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal_2Esup_2E1(V0P_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0P_2E0)) ) ) ).
tff(thm_2Ereal_2Epos__def,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Epos_2E1(V0x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0),V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ).
tff(thm_2Ereal_2Emin__def,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0),V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ereal_2Emax__def,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emax_2E2(V0x_2E0,V1y_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0),V1y_2E0,V0x_2E0) ) ).
tff(thm_2Ereal_2Einf__def,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1r_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0p_2E0),V1r_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,c_2Erealax_2Ereal__neg_2E1(V1r_2E0)) )
=> ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal_2Einf_2E1(V0p_2E0) = c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Esup_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0p_2E0))) ) ) ).
tff(thm_2Ereal_2ENUM__FLOOR__def,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1n_2E0) = c_2Ereal_2Ereal__gt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2B_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2ENUM__FLOOR_2E1(V0x_2E0) = c_2Ewhile_2ELEAST_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0)) ) ) ).
tff(thm_2Ereal_2ENUM__CEILING__def,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1n_2E0) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2ENUM__CEILING_2E1(V0x_2E0) = c_2Ewhile_2ELEAST_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__0,axiom,
c_2Erealax_2Ereal__0_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Ereal_2EREAL__1,axiom,
c_2Erealax_2Ereal__1_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Ereal_2EREAL__10,axiom,
c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Ereal_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_2Ereal_2EREAL__ADD__COMM,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_2Ereal_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_2Ereal_2EREAL__ADD__LID,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__ADD__LINV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ereal_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_2Ereal_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_2Ereal_2EREAL__LT__REFL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V0x_2E0)) ).
tff(thm_2Ereal_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_2Ereal_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_2Ereal_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_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_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_2Ereal_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_2Ereal_2EREAL__MUL__COMM,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_2Ereal_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_2Ereal_2EREAL__MUL__LID,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__MUL__LINV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(V0x_2E0),V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ).
tff(thm_2Ereal_2EREAL__LT__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__INV__0,axiom,
c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Ereal_2EREAL__ADD__RID,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__ADD__RINV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(V0x_2E0,c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ereal_2EREAL__MUL__RID,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__MUL__RINV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,c_2Erealax_2Einv_2E1(V0x_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ).
tff(thm_2Ereal_2EREAL__RDISTRIB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),V2z_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0),c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0)) ) ).
tff(thm_2Ereal_2EREAL__EQ__LADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2z_2E0) )
<=> ( V1y_2E0 = V2z_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__EQ__RADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2z_2E0) = c_2Erealax_2Ereal__add_2E2(V1y_2E0,V2z_2E0) )
<=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__ADD__LID__UNIQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0) = V1y_2E0 )
<=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__ADD__RID__UNIQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0) = V0x_2E0 )
<=> ( V1y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LNEG__UNIQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V0x_2E0 = c_2Erealax_2Ereal__neg_2E1(V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__RNEG__UNIQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V1y_2E0 = c_2Erealax_2Ereal__neg_2E1(V0x_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__NEG__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),c_2Erealax_2Ereal__neg_2E1(V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__MUL__LZERO,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ereal_2EREAL__MUL__RZERO,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ereal_2EREAL__NEG__LMUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),V1y_2E0) ) ).
tff(thm_2Ereal_2EREAL__NEG__RMUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__mul_2E2(V0x_2E0,c_2Erealax_2Ereal__neg_2E1(V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__NEGNEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__NEG__MUL2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),c_2Erealax_2Ereal__neg_2E1(V1y_2E0)) = c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ereal_2EREAL__ENTIRE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V1y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Ereal_2EREAL__LT__LADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2z_2E0)) = c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V2z_2E0) ) ).
tff(thm_2Ereal_2EREAL__LT__RADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2z_2E0),c_2Erealax_2Ereal__add_2E2(V1y_2E0,V2z_2E0)) = c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ereal_2EREAL__NOT__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ~ p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
<=> p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LT__ANTISYM,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
~ ( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LT__GT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
=> ~ p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal_2EREAL__NOT__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ~ p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
<=> p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LE__TOTAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LET__TOTAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
| p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LTE__TOTAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LE__REFL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V0x_2E0)) ).
tff(thm_2Ereal_2EREAL__LE__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
| ( V0x_2E0 = V1y_2E0 ) ) ) ).
tff(thm_2Ereal_2EREAL__LT__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
& ( V0x_2E0 != V1y_2E0 ) ) ) ).
tff(thm_2Ereal_2EREAL__LT__IMP__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LTE__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_2Ereal_2Ereal__lte_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LET__TRANS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_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_2Ereal_2EREAL__LE__TRANS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LE__ANTISYM,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V0x_2E0)) )
<=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__LET__ANTISYM,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
~ ( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LTE__ANTISYM,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
~ ( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LTE__ANTSYM,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
~ ( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal_2EREAL__NEG__LT0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__NEG__GT0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) = c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ).
tff(thm_2Ereal_2EREAL__NEG__LE0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__NEG__GE0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LT__NEGTOTAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
| p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__neg_2E1(V0x_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LE__NEGTOTAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__neg_2E1(V0x_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LE__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LE__SQUARE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V0x_2E0))) ).
tff(thm_2Ereal_2EREAL__LE__01,axiom,
p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ).
tff(thm_2Ereal_2EREAL__LT__01,axiom,
p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ).
tff(thm_2Ereal_2EREAL__LE__LADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2z_2E0)) = c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V2z_2E0) ) ).
tff(thm_2Ereal_2EREAL__LE__RADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2z_2E0),c_2Erealax_2Ereal__add_2E2(V1y_2E0,V2z_2E0)) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ereal_2EREAL__LT__ADD2,axiom,
! [V0w_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V0w_2E0,V1x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2y_2E0,V3z_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(V0w_2E0,V2y_2E0),c_2Erealax_2Ereal__add_2E2(V1x_2E0,V3z_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LE__ADD2,axiom,
! [V0w_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0w_2E0,V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2y_2E0,V3z_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(V0w_2E0,V2y_2E0),c_2Erealax_2Ereal__add_2E2(V1x_2E0,V3z_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LE__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LT__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LT__ADDNEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(V1y_2E0,c_2Erealax_2Ereal__add_2E2(V0x_2E0,c_2Erealax_2Ereal__neg_2E1(V2z_2E0))) = c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(V1y_2E0,V2z_2E0),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__LT__ADDNEG2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,c_2Erealax_2Ereal__neg_2E1(V1y_2E0)),V2z_2E0) = c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Erealax_2Ereal__add_2E2(V2z_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LT__ADD1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Erealax_2Ereal__add_2E2(V1y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) ) ).
tff(thm_2Ereal_2EREAL__SUB__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0),V1y_2E0) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__SUB__ADD2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(V1y_2E0,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__SUB__REFL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ereal_2EREAL__SUB__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__LE__DOUBLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V0x_2E0)) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__LE__NEGL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),V0x_2E0) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__LE__NEGR,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ).
tff(thm_2Ereal_2EREAL__NEG__EQ0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg_2E1(V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__NEG__0,axiom,
c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Ereal_2EREAL__NEG__SUB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__SUB__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__SUB__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__ADD__SUB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),V0x_2E0) = V1y_2E0 ) ).
tff(thm_2Ereal_2EREAL__EQ__LMUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0) )
<=> ( ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V1y_2E0 = V2z_2E0 ) ) ) ).
tff(thm_2Ereal_2EREAL__EQ__RMUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0) = c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0) )
<=> ( ( V2z_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V0x_2E0 = V1y_2E0 ) ) ) ).
tff(thm_2Ereal_2EREAL__SUB__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_2Ereal_2Ereal__sub_2E2(V1y_2E0,V2z_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Ereal_2EREAL__SUB__RDISTRIB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0),V2z_2E0) = c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0),c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0)) ) ).
tff(thm_2Ereal_2EREAL__NEG__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg_2E1(V0x_2E0) = V1y_2E0 )
<=> ( V0x_2E0 = c_2Erealax_2Ereal__neg_2E1(V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__NEG__MINUS1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(V0x_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__INV__NZ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Erealax_2Einv_2E1(V0x_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__INVINV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Erealax_2Einv_2E1(c_2Erealax_2Einv_2E1(V0x_2E0)) = V0x_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__LT__IMP__NE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
=> ( V0x_2E0 != V1y_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__INV__POS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Einv_2E1(V0x_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LT__LMUL__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LT__RMUL__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LT__LMUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0)) = c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V2z_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LT__RMUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0),c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0)) = c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LT__RMUL__IMP,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(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0),c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LT__LMUL__IMP,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_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LINV__UNIQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
=> ( V0x_2E0 = c_2Erealax_2Einv_2E1(V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__RINV__UNIQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
=> ( V1y_2E0 = c_2Erealax_2Einv_2E1(V0x_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__INV__INV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Einv_2E1(c_2Erealax_2Einv_2E1(V0x_2E0)) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__INV__EQ__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Einv_2E1(V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__NEG__INV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Einv_2E1(V0x_2E0)) = c_2Erealax_2Einv_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__INV__1OVER,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Einv_2E1(V0x_2E0) = c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__LT__INV__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Einv_2E1(V0x_2E0)) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__LE__INV__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Einv_2E1(V0x_2E0)) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__LE__INV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Einv_2E1(V0x_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LE__ADDR,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0) ) ).
tff(thm_2Ereal_2EREAL__LE__ADDL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(V1y_2E0,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__LT__ADDR,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0) ) ).
tff(thm_2Ereal_2EREAL__LT__ADDL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(V1y_2E0,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ).
tff(thm_2Ereal_2EREAL__POS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(V0n_2E0))) ).
tff(thm_2Ereal_2EREAL__LE,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(V0m_2E0),c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0) ) ).
tff(thm_2Ereal_2EREAL__LT,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(V0m_2E0),c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)) = c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0) ) ).
tff(thm_2Ereal_2EREAL__INJ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__of__num_2E1(V0m_2E0) = c_2Ereal_2Ereal__of__num_2E1(V1n_2E0) )
<=> ( V0m_2E0 = V1n_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(V0m_2E0),c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Ereal_2EREAL__MUL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V0m_2E0),c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Ereal_2EREAL__INV1,axiom,
c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Ereal_2EREAL__OVER1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__DIV__REFL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal_2E_2F_2E2(V0x_2E0,V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ).
tff(thm_2Ereal_2EREAL__DIV__LZERO,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ereal_2EREAL__LT__NZ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__of__num_2E1(V0n_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(V0n_2E0))) ) ).
tff(thm_2Ereal_2EREAL__NZ__IMP__LT,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( ( V0n_2E0 != c_2Enum_2E0_2E0 )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(V0n_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LT__RDIV__0,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1z_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(V0y_2E0,V1z_2E0)) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LT__RDIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,V2z_2E0),c_2Ereal_2E_2F_2E2(V1y_2E0,V2z_2E0)) = c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LT__FRACTION__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1d_2E0: tyop_2Erealax_2Ereal] :
( ( V0n_2E0 != c_2Enum_2E0_2E0 )
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(V1d_2E0,c_2Ereal_2Ereal__of__num_2E1(V0n_2E0))) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1d_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LT__MULTIPLE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(V1d_2E0,c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),V1d_2E0)) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1d_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LT__FRACTION,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(V1d_2E0,c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)),V1d_2E0) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1d_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LT__HALF1,axiom,
! [V0d_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(V0d_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0d_2E0) ) ).
tff(thm_2Ereal_2EREAL__LT__HALF2,axiom,
! [V0d_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(V0d_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),V0d_2E0) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0d_2E0) ) ).
tff(thm_2Ereal_2EREAL__DOUBLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(V0x_2E0,V0x_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__DIV__LMUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( V1y_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Erealax_2Ereal__mul_2E2(V1y_2E0,c_2Ereal_2E_2F_2E2(V0x_2E0,V1y_2E0)) = V0x_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__DIV__RMUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( V1y_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,V1y_2E0),V1y_2E0) = V0x_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__HALF__DOUBLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__DOWN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ? [V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__DOWN2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> ? [V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,V1y_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__SUB__SUB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0),V0x_2E0) = c_2Erealax_2Ereal__neg_2E1(V1y_2E0) ) ).
tff(thm_2Ereal_2EREAL__LT__ADD__SUB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),V2z_2E0) = c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__sub_2E2(V2z_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LT__SUB__RADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0),V2z_2E0) = c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Erealax_2Ereal__add_2E2(V2z_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LT__SUB__LADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V2z_2E0)) = c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2z_2E0),V1y_2E0) ) ).
tff(thm_2Ereal_2EREAL__LE__SUB__LADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V2z_2E0)) = c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2z_2E0),V1y_2E0) ) ).
tff(thm_2Ereal_2EREAL__LE__SUB__RADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0),V2z_2E0) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Erealax_2Ereal__add_2E2(V2z_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LT__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),c_2Erealax_2Ereal__neg_2E1(V1y_2E0)) = c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__LE__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),c_2Erealax_2Ereal__neg_2E1(V1y_2E0)) = c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__ADD2__SUB2,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__add_2E2(V0a_2E0,V1b_2E0),c_2Erealax_2Ereal__add_2E2(V2c_2E0,V3d_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__sub_2E2(V0a_2E0,V2c_2E0),c_2Ereal_2Ereal__sub_2E2(V1b_2E0,V3d_2E0)) ) ).
tff(thm_2Ereal_2EREAL__SUB__LZERO,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) = c_2Erealax_2Ereal__neg_2E1(V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__SUB__RZERO,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__LET__ADD2,axiom,
! [V0w_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0w_2E0,V1x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2y_2E0,V3z_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(V0w_2E0,V2y_2E0),c_2Erealax_2Ereal__add_2E2(V1x_2E0,V3z_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LTE__ADD2,axiom,
! [V0w_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V0w_2E0,V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2y_2E0,V3z_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(V0w_2E0,V2y_2E0),c_2Erealax_2Ereal__add_2E2(V1x_2E0,V3z_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LET__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LTE__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LT__MUL2,axiom,
! [V0x1_2E0: tyop_2Erealax_2Ereal,V1x2_2E0: tyop_2Erealax_2Ereal,V2y1_2E0: tyop_2Erealax_2Ereal,V3y2_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x1_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2y1_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x1_2E0,V1x2_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2y1_2E0,V3y2_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0x1_2E0,V2y1_2E0),c_2Erealax_2Ereal__mul_2E2(V1x2_2E0,V3y2_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LT__INV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Einv_2E1(V1y_2E0),c_2Erealax_2Einv_2E1(V0x_2E0))) ) ).
tff(thm_2Ereal_2EREAL__SUB__LNEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),V1y_2E0) = c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__SUB__RNEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(V0x_2E0,c_2Erealax_2Ereal__neg_2E1(V1y_2E0)) = c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ereal_2EREAL__SUB__NEG2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),c_2Erealax_2Ereal__neg_2E1(V1y_2E0)) = c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__SUB__TRIANGLE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__sub_2E2(V0a_2E0,V1b_2E0),c_2Ereal_2Ereal__sub_2E2(V1b_2E0,V2c_2E0)) = c_2Ereal_2Ereal__sub_2E2(V0a_2E0,V2c_2E0) ) ).
tff(thm_2Ereal_2EREAL__EQ__SUB__LADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 = c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V2z_2E0) )
<=> ( c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2z_2E0) = V1y_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__EQ__SUB__RADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0) = V2z_2E0 )
<=> ( V0x_2E0 = c_2Erealax_2Ereal__add_2E2(V2z_2E0,V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__INV__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V1y_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Erealax_2Einv_2E1(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(V0x_2E0),c_2Erealax_2Einv_2E1(V1y_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__LE__LMUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0)) = c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V2z_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LE__RMUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
=> ( c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0),c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0)) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__SUB__INV2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V1y_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Einv_2E1(V0x_2E0),c_2Erealax_2Einv_2E1(V1y_2E0)) = c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V0x_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__SUB__SUB2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(V0x_2E0,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)) = V1y_2E0 ) ).
tff(thm_2Ereal_2EREAL__ADD__SUB2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(V0x_2E0,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__neg_2E1(V1y_2E0) ) ).
tff(thm_2Ereal_2EREAL__MEAN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
=> ? [V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V2z_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,V1y_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__EQ__LMUL2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( ( V1y_2E0 = V2z_2E0 )
<=> ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0) ) ) ) ).
tff(thm_2Ereal_2EREAL__LE__MUL2,axiom,
! [V0x1_2E0: tyop_2Erealax_2Ereal,V1x2_2E0: tyop_2Erealax_2Ereal,V2y1_2E0: tyop_2Erealax_2Ereal,V3y2_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x1_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2y1_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x1_2E0,V1x2_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2y1_2E0,V3y2_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0x1_2E0,V2y1_2E0),c_2Erealax_2Ereal__mul_2E2(V1x2_2E0,V3y2_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LE__LDIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,c_2Erealax_2Ereal__mul_2E2(V2z_2E0,V0x_2E0))) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2E_2F_2E2(V1y_2E0,V0x_2E0),V2z_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LE__RDIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V0x_2E0),V2z_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,c_2Ereal_2E_2F_2E2(V2z_2E0,V0x_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LT__DIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LE__DIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LT__1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,V1y_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ereal_2EREAL__LE__LMUL__IMP,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LE__RMUL__IMP,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V0x_2E0),c_2Erealax_2Ereal__mul_2E2(V2z_2E0,V0x_2E0))) ) ).
tff(thm_2Ereal_2EREAL__EQ__IMP__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 = V1y_2E0 )
=> p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__INV__LT1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Erealax_2Einv_2E1(V0x_2E0))) ) ).
tff(thm_2Ereal_2EREAL__POS__NZ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__EQ__RMUL__IMP,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( ( V2z_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0) = c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0) ) )
=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__EQ__LMUL__IMP,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0) ) )
=> ( V1y_2E0 = V2z_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__FACT__NZ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V0n_2E0)) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ereal_2EREAL__DIFFSQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V0x_2E0),c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__POASQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V0x_2E0)))
<=> ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__SUMSQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V0x_2E0),c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V1y_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V1y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Ereal_2EREAL__EQ__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg_2E1(V0x_2E0) = c_2Erealax_2Ereal__neg_2E1(V1y_2E0) )
<=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__DIV__MUL2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V1z_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ! [V2y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(V2y_2E0,V1z_2E0) = c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2y_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1z_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__MIDDLE1,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V1b_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__add_2E2(V0a_2E0,V1b_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))) ) ).
tff(thm_2Ereal_2EREAL__MIDDLE2,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V1b_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__add_2E2(V0a_2E0,V1b_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),V1b_2E0)) ) ).
tff(thm_2Ereal_2EABS__ZERO,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs_2E1(V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2EABS__0,axiom,
c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Ereal_2EABS__1,axiom,
c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Ereal_2EABS__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) = c_2Ereal_2Eabs_2E1(V0x_2E0) ) ).
tff(thm_2Ereal_2EABS__TRIANGLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)))) ).
tff(thm_2Ereal_2EABS__TRIANGLE__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)))) ).
tff(thm_2Ereal_2EABS__TRIANGLE__SUB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Eabs_2E1(V1y_2E0),c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0))))) ).
tff(thm_2Ereal_2EABS__TRIANGLE__LT,axiom,
! [V0e_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Eabs_2E1(V1x_2E0),c_2Ereal_2Eabs_2E1(V2y_2E0)),V0e_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Erealax_2Ereal__add_2E2(V1x_2E0,V2y_2E0)),V0e_2E0)) ) ).
tff(thm_2Ereal_2EABS__POS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Eabs_2E1(V0x_2E0))) ).
tff(thm_2Ereal_2EABS__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)) ) ).
tff(thm_2Ereal_2EABS__LT__MUL2,axiom,
! [V0w_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(V0w_2E0),V2y_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(V1x_2E0),V3z_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Erealax_2Ereal__mul_2E2(V0w_2E0,V1x_2E0)),c_2Erealax_2Ereal__mul_2E2(V2y_2E0,V3z_2E0))) ) ).
tff(thm_2Ereal_2EABS__SUB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal_2EABS__NZ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Eabs_2E1(V0x_2E0))) ) ).
tff(thm_2Ereal_2EABS__INV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal_2Eabs_2E1(c_2Erealax_2Einv_2E1(V0x_2E0)) = c_2Erealax_2Einv_2E1(c_2Ereal_2Eabs_2E1(V0x_2E0)) ) ) ).
tff(thm_2Ereal_2EABS__ABS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(c_2Ereal_2Eabs_2E1(V0x_2E0)) = c_2Ereal_2Eabs_2E1(V0x_2E0) ) ).
tff(thm_2Ereal_2EABS__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Eabs_2E1(V0x_2E0))) ).
tff(thm_2Ereal_2EABS__REFL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs_2E1(V0x_2E0) = V0x_2E0 )
<=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0)) ) ).
tff(thm_2Ereal_2EABS__N,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)) = c_2Ereal_2Ereal__of__num_2E1(V0n_2E0) ) ).
tff(thm_2Ereal_2EABS__BETWEEN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2d_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V2d_2E0),V1y_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2d_2E0))) )
<=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V0x_2E0)),V2d_2E0)) ) ).
tff(thm_2Ereal_2EABS__BOUND,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)),V2d_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2d_2E0))) ) ).
tff(thm_2Ereal_2EABS__STILLNZ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)),c_2Ereal_2Eabs_2E1(V1y_2E0)))
=> ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2EABS__CASES,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Eabs_2E1(V0x_2E0))) ) ).
tff(thm_2Ereal_2EABS__BETWEEN1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V2z_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V0x_2E0)),c_2Ereal_2Ereal__sub_2E2(V2z_2E0,V0x_2E0))) )
=> p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V2z_2E0)) ) ).
tff(thm_2Ereal_2EABS__SIGN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)),V1y_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0)) ) ).
tff(thm_2Ereal_2EABS__SIGN2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)),c_2Erealax_2Ereal__neg_2E1(V1y_2E0)))
=> p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Ereal_2EABS__DIV,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( ( V0y_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0)) = c_2Ereal_2E_2F_2E2(c_2Ereal_2Eabs_2E1(V1x_2E0),c_2Ereal_2Eabs_2E1(V0y_2E0)) ) ) ).
tff(thm_2Ereal_2EABS__CIRCLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2h_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(V2h_2E0),c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Eabs_2E1(V1y_2E0),c_2Ereal_2Eabs_2E1(V0x_2E0))))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2h_2E0)),c_2Ereal_2Eabs_2E1(V1y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__SUB__ABS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)),c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)))) ).
tff(thm_2Ereal_2EABS__SUB__ABS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0))),c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)))) ).
tff(thm_2Ereal_2EABS__BETWEEN2,axiom,
! [V0x0_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y0_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V0x0_2E0,V2y0_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1x_2E0,V0x0_2E0)),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__sub_2E2(V2y0_2E0,V0x0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V3y_2E0,V2y0_2E0)),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__sub_2E2(V2y0_2E0,V0x0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))) )
=> p(c_2Erealax_2Ereal__lt_2E2(V1x_2E0,V3y_2E0)) ) ).
tff(thm_2Ereal_2EABS__BOUNDS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1k_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),V1k_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__neg_2E1(V1k_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1k_2E0)) ) ) ).
tff(thm_2Ereal_2EPOW__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Enum_2ESUC_2E1(V0n_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ereal_2EPOW__NZ,axiom,
! [V0c_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
( ( V0c_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal_2Epow_2E2(V0c_2E0,V1n_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2EPOW__INV,axiom,
! [V0c_2E0: tyop_2Erealax_2Ereal] :
( ( V0c_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ! [V1n_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Einv_2E1(c_2Ereal_2Epow_2E2(V0c_2E0,V1n_2E0)) = c_2Ereal_2Epow_2E2(c_2Erealax_2Einv_2E1(V0c_2E0),V1n_2E0) ) ) ).
tff(thm_2Ereal_2EPOW__ABS,axiom,
! [V0c_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(c_2Ereal_2Eabs_2E1(V0c_2E0),V1n_2E0) = c_2Ereal_2Eabs_2E1(c_2Ereal_2Epow_2E2(V0c_2E0,V1n_2E0)) ) ).
tff(thm_2Ereal_2EPOW__PLUS1,axiom,
! [V0e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0e_2E0))
=> ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0),V0e_2E0)),c_2Ereal_2Epow_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0e_2E0),V1n_2E0))) ) ).
tff(thm_2Ereal_2EPOW__ADD,axiom,
! [V0c_2E0: tyop_2Erealax_2Ereal,V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(V0c_2E0,c_2Earithmetic_2E_2B_2E2(V1m_2E0,V2n_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Epow_2E2(V0c_2E0,V1m_2E0),c_2Ereal_2Epow_2E2(V0c_2E0,V2n_2E0)) ) ).
tff(thm_2Ereal_2EPOW__1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) = V0x_2E0 ) ).
tff(thm_2Ereal_2EPOW__2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V0x_2E0) ) ).
tff(thm_2Ereal_2EPOW__ONE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0n_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
tff(thm_2Ereal_2EPOW__POS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0))) ) ).
tff(thm_2Ereal_2EPOW__LE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V2y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Epow_2E2(V1x_2E0,V0n_2E0),c_2Ereal_2Epow_2E2(V2y_2E0,V0n_2E0))) ) ).
tff(thm_2Ereal_2EPOW__M1,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs_2E1(c_2Ereal_2Epow_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0n_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
tff(thm_2Ereal_2EPOW__MUL,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Epow_2E2(c_2Erealax_2Ereal__mul_2E2(V1x_2E0,V2y_2E0),V0n_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Epow_2E2(V1x_2E0,V0n_2E0),c_2Ereal_2Epow_2E2(V2y_2E0,V0n_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LE__POW2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ).
tff(thm_2Ereal_2EABS__POW2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
tff(thm_2Ereal_2EREAL__POW2__ABS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Epow_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LE1__POW2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ereal_2EREAL__LT1__POW2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ereal_2EPOW__POS__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Epow_2E2(V0x_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)))) ) ).
tff(thm_2Ereal_2EPOW__2__LE1,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0n_2E0))) ).
tff(thm_2Ereal_2EPOW__2__LT,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0n_2E0))) ).
tff(thm_2Ereal_2EPOW__MINUS1,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
tff(thm_2Ereal_2EPOW__LT,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1x_2E0,V2y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Epow_2E2(V1x_2E0,c_2Enum_2ESUC_2E1(V0n_2E0)),c_2Ereal_2Epow_2E2(V2y_2E0,c_2Enum_2ESUC_2E1(V0n_2E0)))) ) ).
tff(thm_2Ereal_2EREAL__POW__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0))) ) ).
tff(thm_2Ereal_2EPOW__EQ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2y_2E0))
& ( c_2Ereal_2Epow_2E2(V1x_2E0,c_2Enum_2ESUC_2E1(V0n_2E0)) = c_2Ereal_2Epow_2E2(V2y_2E0,c_2Enum_2ESUC_2E1(V0n_2E0)) ) )
=> ( V1x_2E0 = V2y_2E0 ) ) ).
tff(thm_2Ereal_2EPOW__ZERO,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epow_2E2(V1x_2E0,V0n_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( V1x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2EPOW__ZERO__EQ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epow_2E2(V1x_2E0,c_2Enum_2ESUC_2E1(V0n_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V1x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__POW__LT2,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0n_2E0 != c_2Enum_2E0_2E0 )
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1x_2E0,V2y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Epow_2E2(V1x_2E0,V0n_2E0),c_2Ereal_2Epow_2E2(V2y_2E0,V0n_2E0))) ) ).
tff(thm_2Ereal_2EREAL__POW__MONO__LT,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V2x_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Epow_2E2(V2x_2E0,V0m_2E0),c_2Ereal_2Epow_2E2(V2x_2E0,V1n_2E0))) ) ).
tff(thm_2Ereal_2EREAL__POW__POW,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,V1m_2E0),V2n_2E0) = c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2E_2A_2E2(V1m_2E0,V2n_2E0)) ) ).
tff(thm_2Ereal_2EREAL__SUP__SOMEPOS,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_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0)) )
& ? [V2z_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(V3x_2E0,V2z_2E0)) ) )
=> ? [V4s_2E0: tyop_2Erealax_2Ereal] :
! [V5y_2E0: tyop_2Erealax_2Ereal] :
( ? [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V6x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V5y_2E0,V6x_2E0)) )
<=> p(c_2Erealax_2Ereal__lt_2E2(V5y_2E0,V4s_2E0)) ) ) ).
tff(thm_2Ereal_2ESUP__LEMMA1,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Erealax_2Ereal,V2d_2E0: tyop_2Erealax_2Ereal] :
( ! [V3y_2E0: tyop_2Erealax_2Ereal] :
( ? [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Erealax_2Ereal__add_2E2(V4x_2E0,V2d_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(V3y_2E0,V4x_2E0)) )
<=> p(c_2Erealax_2Ereal__lt_2E2(V3y_2E0,V1s_2E0)) )
=> ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( ? [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V6x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V5y_2E0,V6x_2E0)) )
<=> p(c_2Erealax_2Ereal__lt_2E2(V5y_2E0,c_2Erealax_2Ereal__add_2E2(V1s_2E0,V2d_2E0))) ) ) ).
tff(thm_2Ereal_2ESUP__LEMMA2,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))
=> ? [V2d_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Erealax_2Ereal__add_2E2(V3x_2E0,V2d_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3x_2E0)) ) ) ).
tff(thm_2Ereal_2ESUP__LEMMA3,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1d_2E0: tyop_2Erealax_2Ereal] :
( ? [V2z_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(V3x_2E0,V2z_2E0)) )
=> ? [V4z_2E0: tyop_2Erealax_2Ereal] :
! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Erealax_2Ereal__add_2E2(V5x_2E0,V1d_2E0)))
=> p(c_2Erealax_2Ereal__lt_2E2(V5x_2E0,V4z_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__SUP__EXISTS,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))
& ? [V2z_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(V3x_2E0,V2z_2E0)) ) )
=> ? [V4s_2E0: tyop_2Erealax_2Ereal] :
! [V5y_2E0: tyop_2Erealax_2Ereal] :
( ? [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V6x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V5y_2E0,V6x_2E0)) )
<=> p(c_2Erealax_2Ereal__lt_2E2(V5y_2E0,V4s_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__SUP,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))
& ? [V2z_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(V3x_2E0,V2z_2E0)) ) )
=> ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( ? [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V4y_2E0,V5x_2E0)) )
<=> p(c_2Erealax_2Ereal__lt_2E2(V4y_2E0,c_2Ereal_2Esup_2E1(V0P_2E0))) ) ) ).
tff(thm_2Ereal_2EREAL__SUP__UBOUND,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))
& ? [V2z_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(V3x_2E0,V2z_2E0)) ) )
=> ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V4y_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V4y_2E0,c_2Ereal_2Esup_2E1(V0P_2E0))) ) ) ).
tff(thm_2Ereal_2ESETOK__LE__LT,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))
& ? [V2z_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2z_2E0)) ) )
<=> ( ? [V4x_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V4x_2E0))
& ? [V5z_2E0: tyop_2Erealax_2Ereal] :
! [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V6x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(V6x_2E0,V5z_2E0)) ) ) ) ).
tff(thm_2Ereal_2EREAL__SUP__LE,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))
& ? [V2z_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2z_2E0)) ) )
=> ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( ? [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V4y_2E0,V5x_2E0)) )
<=> p(c_2Erealax_2Ereal__lt_2E2(V4y_2E0,c_2Ereal_2Esup_2E1(V0P_2E0))) ) ) ).
tff(thm_2Ereal_2EREAL__SUP__UBOUND__LE,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))
& ? [V2z_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2z_2E0)) ) )
=> ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V4y_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V4y_2E0,c_2Ereal_2Esup_2E1(V0P_2E0))) ) ) ).
tff(thm_2Ereal_2EREAL__ARCH,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ! [V1y_2E0: tyop_2Erealax_2Ereal] :
? [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V2n_2E0),V0x_2E0))) ) ).
tff(thm_2Ereal_2EREAL__ARCH__LEAST,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0y_2E0))
=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
=> ? [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V2n_2E0),V0y_2E0),V1x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1x_2E0,c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2ESUC_2E1(V2n_2E0)),V0y_2E0))) ) ) ) ).
tff(thm_2Ereal_2Esum__ind,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool))] :
( ( ! [V1n_2E0: tyop_2Enum_2Enum,V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : p(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1n_2E0,c_2Enum_2E0_2E0)),V2f_2E0))
& ! [V3n_2E0: tyop_2Enum_2Enum,V4m_2E0: tyop_2Enum_2Enum,V5f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V3n_2E0,V4m_2E0)),V5f_2E0))
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V3n_2E0,c_2Enum_2ESUC_2E1(V4m_2E0))),V5f_2E0)) ) )
=> ! [V6v_2E0: tyop_2Enum_2Enum,V7v1_2E0: tyop_2Enum_2Enum,V8v2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : p(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V6v_2E0,V7v1_2E0)),V8v2_2E0)) ) ).
tff(thm_2Ereal_2Esum,axiom,
( ! [V0n_2E0: tyop_2Enum_2Enum,V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0n_2E0,c_2Enum_2E0_2E0),V1f_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ! [V2n_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum,V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2n_2E0,c_2Enum_2ESUC_2E1(V3m_2E0)),V4f_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2n_2E0,V3m_2E0),V4f_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V4f_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,V3m_2E0))) ) ) ).
tff(thm_2Ereal_2Esum__compute,axiom,
( ! [V0n_2E0: tyop_2Enum_2Enum,V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0n_2E0,c_2Enum_2E0_2E0),V1f_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ! [V2n_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum,V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V3m_2E0))),V4f_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2n_2E0,c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V3m_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V4f_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V4f_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V3m_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) )
& ! [V5n_2E0: tyop_2Enum_2Enum,V6m_2E0: tyop_2Enum_2Enum,V7f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V5n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V6m_2E0))),V7f_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V5n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V6m_2E0))),V7f_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V7f_2E0,c_2Earithmetic_2E_2B_2E2(V5n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V6m_2E0))))) ) ) ).
tff(thm_2Ereal_2ESUM__TWO,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V1n_2E0),V0f_2E0),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1n_2E0,V2p_2E0),V0f_2E0)) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)),V0f_2E0) ) ).
tff(thm_2Ereal_2ESUM__DIFF,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1m_2E0,V2n_2E0),V0f_2E0) = c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,c_2Earithmetic_2E_2B_2E2(V1m_2E0,V2n_2E0)),V0f_2E0),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V1m_2E0),V0f_2E0)) ) ).
tff(thm_2Ereal_2EABS__SUM,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V3n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1m_2E0,V2n_2E0),V0f_2E0)),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1m_2E0,V2n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)))) ) ).
tff(thm_2Ereal_2ESUM__LE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( ! [V4r_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V4r_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V4r_2E0,c_2Earithmetic_2E_2B_2E2(V3n_2E0,V2m_2E0))) )
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4r_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4r_2E0))) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V0f_2E0),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V1g_2E0))) ) ).
tff(thm_2Ereal_2ESUM__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( ! [V4r_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V4r_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V4r_2E0,c_2Earithmetic_2E_2B_2E2(V3n_2E0,V2m_2E0))) )
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4r_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4r_2E0) ) )
=> ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V0f_2E0) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V1g_2E0) ) ) ).
tff(thm_2Ereal_2ESUM__POS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0)))
=> ! [V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V0f_2E0))) ) ).
tff(thm_2Ereal_2ESUM__POS__GEN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0))) )
=> ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1m_2E0,V3n_2E0),V0f_2E0))) ) ).
tff(thm_2Ereal_2ESUM__ABS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F1_2E0,V0f_2E0),V4m_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4m_2E0)) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V3m_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3m_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs_2E1(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1m_2E0,V2n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0))) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1m_2E0,V2n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F1_2E0,V0f_2E0)) ) ) ) ).
tff(thm_2Ereal_2ESUM__ABS__LE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V3n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1m_2E0,V2n_2E0),V0f_2E0)),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1m_2E0,V2n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)))) ) ).
tff(thm_2Ereal_2ESUM__ZERO,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1N_2E0: tyop_2Enum_2Enum] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V2n_2E0,V1N_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V3m_2E0,V1N_2E0))
=> ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V3m_2E0,V4n_2E0),V0f_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Ereal_2ESUM__ADD,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V4n_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V0f_2E0),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V1g_2E0)) ) ) ).
tff(thm_2Ereal_2ESUM__CMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0f_2E0),V4n_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal,V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0f_2E0)) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V0f_2E0)) ) ) ).
tff(thm_2Ereal_2ESUM__NEG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V3n_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum,V2d_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1n_2E0,V2d_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) = c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1n_2E0,V2d_2E0),V0f_2E0)) ) ) ).
tff(thm_2Ereal_2ESUM__SUB,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V4n_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V0f_2E0),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V1g_2E0)) ) ) ).
tff(thm_2Ereal_2ESUM__SUBST,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( ! [V4p_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V4p_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V4p_2E0,c_2Earithmetic_2E_2B_2E2(V2m_2E0,V3n_2E0))) )
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4p_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4p_2E0) ) )
=> ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V0f_2E0) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V1g_2E0) ) ) ).
tff(thm_2Ereal_2ESUM__NSUB,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2c_2E0: tyop_2Erealax_2Ereal,V3p_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2c_2E0),V3p_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1f_2E0,V3p_2E0),V2c_2E0) )
=> ! [V0n_2E0: tyop_2Enum_2Enum,V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2c_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V0n_2E0),V1f_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),V2c_2E0)) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V0n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2c_2E0)) ) ) ).
tff(thm_2Ereal_2ESUM__BOUND,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Erealax_2Ereal,V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( ! [V4p_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V4p_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V4p_2E0,c_2Earithmetic_2E_2B_2E2(V2m_2E0,V3n_2E0))) )
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4p_2E0),V1k_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V0f_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V3n_2E0),V1k_2E0))) ) ).
tff(thm_2Ereal_2ESUM__GROUP,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1k_2E0: tyop_2Enum_2Enum,V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1k_2E0),V2f_2E0),V3m_2E0) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2E_2A_2E2(V3m_2E0,V1k_2E0),V1k_2E0),V2f_2E0) )
=> ! [V0n_2E0: tyop_2Enum_2Enum,V1k_2E0: tyop_2Enum_2Enum,V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V0n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1k_2E0),V2f_2E0)) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,c_2Earithmetic_2E_2A_2E2(V0n_2E0,V1k_2E0)),V2f_2E0) ) ) ).
tff(thm_2Ereal_2ESUM__1,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0f_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0) ) ).
tff(thm_2Ereal_2ESUM__2,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0f_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ereal_2ESUM__OFFSET,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2k_2E0),V3m_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V3m_2E0,V2k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum,V2k_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V1n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2k_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2k_2E0)),V0f_2E0),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V2k_2E0),V0f_2E0)) ) ) ).
tff(thm_2Ereal_2ESUM__REINDEX,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Enum_2Enum,V4r_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2k_2E0),V4r_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V4r_2E0,V2k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2k_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(V1m_2E0,V2k_2E0),V3n_2E0),V0f_2E0) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1m_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2k_2E0)) ) ) ).
tff(thm_2Ereal_2ESUM__0,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V2r_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,F0_2E0,V2r_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0m_2E0,V1n_2E0),F0_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal_2ESUM__PERMUTE__0,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0n_2E0: tyop_2Enum_2Enum,V1p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2y_2E0: tyop_2Enum_2Enum,V3x_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0,V0n_2E0),V1p_2E0),V2y_2E0),V3x_2E0))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V3x_2E0,V0n_2E0))
& ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1p_2E0,V3x_2E0) = V2y_2E0 ) ) )
=> ( ! [V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F1_2E0,V4f_2E0),V1p_2E0),V5n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V4f_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1p_2E0,V5n_2E0)) )
=> ! [V0n_2E0: tyop_2Enum_2Enum,V1p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( ! [V2y_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V2y_2E0,V0n_2E0))
=> p(c_2Ebool_2E_3F_21_2E1(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0,V0n_2E0),V1p_2E0),V2y_2E0))) )
=> ! [V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V0n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F1_2E0,V4f_2E0),V1p_2E0)) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V0n_2E0),V4f_2E0) ) ) ) ) ).
tff(thm_2Ereal_2ESUM__CANCEL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V3n_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum,V2d_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1n_2E0,V2d_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2d_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__MUL__RNEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,c_2Erealax_2Ereal__neg_2E1(V1y_2E0)) = c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__MUL__LNEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),V1y_2E0) = c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal_2Ereal__lt,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V1x_2E0,V0y_2E0))
<=> ~ p(c_2Ereal_2Ereal__lte_2E2(V0y_2E0,V1x_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LE__LADD__IMP,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V2z_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2z_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LE__LNEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),V1y_2E0) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LE__NEG2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),c_2Erealax_2Ereal__neg_2E1(V1y_2E0)) = c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__NEG__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__LE__RNEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Erealax_2Ereal__neg_2E1(V1y_2E0)) = c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ).
tff(thm_2Ereal_2EREAL__POW__INV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(c_2Erealax_2Einv_2E1(V0x_2E0),V1n_2E0) = c_2Erealax_2Einv_2E1(c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0)) ) ).
tff(thm_2Ereal_2EREAL__POW__DIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,V1y_2E0),V2n_2E0) = c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,V2n_2E0),c_2Ereal_2Epow_2E2(V1y_2E0,V2n_2E0)) ) ).
tff(thm_2Ereal_2EREAL__POW__ADD,axiom,
! [V0c_2E0: tyop_2Erealax_2Ereal,V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(V0c_2E0,c_2Earithmetic_2E_2B_2E2(V1m_2E0,V2n_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Epow_2E2(V0c_2E0,V1m_2E0),c_2Ereal_2Epow_2E2(V0c_2E0,V2n_2E0)) ) ).
tff(thm_2Ereal_2EREAL__LE__RDIV__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
=> ( c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2E_2F_2E2(V1y_2E0,V2z_2E0)) = c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0),V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LE__LDIV__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
=> ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,V2z_2E0),V1y_2E0) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__LT__RDIV__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2E_2F_2E2(V1y_2E0,V2z_2E0)) = c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0),V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__LT__LDIV__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,V2z_2E0),V1y_2E0) = c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__EQ__RDIV__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
=> ( ( V0x_2E0 = c_2Ereal_2E_2F_2E2(V1y_2E0,V2z_2E0) )
<=> ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0) = V1y_2E0 ) ) ) ).
tff(thm_2Ereal_2EREAL__EQ__LDIV__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
=> ( ( c_2Ereal_2E_2F_2E2(V0x_2E0,V2z_2E0) = V1y_2E0 )
<=> ( V0x_2E0 = c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0) ) ) ) ).
tff(thm_2Ereal_2EREAL__OF__NUM__POW,axiom,
! [V0x_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(V0x_2E0),V1n_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EEXP_2E2(V0x_2E0,V1n_2E0)) ) ).
tff(thm_2Ereal_2EREAL__ADD__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_2Ereal_2EREAL__ADD__RDISTRIB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),V2z_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0),c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0)) ) ).
tff(thm_2Ereal_2EREAL__OF__NUM__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(V0m_2E0),c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Ereal_2EREAL__OF__NUM__LE,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(V0m_2E0),c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0) ) ).
tff(thm_2Ereal_2EREAL__OF__NUM__MUL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V0m_2E0),c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Ereal_2EREAL__OF__NUM__SUC,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)) ) ).
tff(thm_2Ereal_2EREAL__OF__NUM__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__of__num_2E1(V0m_2E0) = c_2Ereal_2Ereal__of__num_2E1(V1n_2E0) )
<=> ( V0m_2E0 = V1n_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__EQ__MUL__LCANCEL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0) )
<=> ( ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V1y_2E0 = V2z_2E0 ) ) ) ).
tff(thm_2Ereal_2EREAL__ABS__0,axiom,
c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Ereal_2EREAL__ABS__TRIANGLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)))) ).
tff(thm_2Ereal_2EREAL__ABS__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__ABS__POS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Eabs_2E1(V0x_2E0))) ).
tff(thm_2Ereal_2EREAL__LE__EPSILON,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Erealax_2Ereal__add_2E2(V1y_2E0,V2e_2E0))) )
=> p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal_2EREAL__BIGNUM,axiom,
! [V0r_2E0: tyop_2Erealax_2Ereal] :
? [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(V0r_2E0,c_2Ereal_2Ereal__of__num_2E1(V1n_2E0))) ).
tff(thm_2Ereal_2EREAL__INV__LT__ANTIMONO,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Einv_2E1(V0x_2E0),c_2Erealax_2Einv_2E1(V1y_2E0)) = c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V0x_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__INV__INJ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Einv_2E1(V0x_2E0) = c_2Erealax_2Einv_2E1(V1y_2E0) )
<=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__DIV__RMUL__CANCEL,axiom,
! [V0c_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ( V0c_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__mul_2E2(V1a_2E0,V0c_2E0),c_2Erealax_2Ereal__mul_2E2(V2b_2E0,V0c_2E0)) = c_2Ereal_2E_2F_2E2(V1a_2E0,V2b_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__DIV__LMUL__CANCEL,axiom,
! [V0c_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ( V0c_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__mul_2E2(V0c_2E0,V1a_2E0),c_2Erealax_2Ereal__mul_2E2(V0c_2E0,V2b_2E0)) = c_2Ereal_2E_2F_2E2(V1a_2E0,V2b_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__DIV__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2E_2F_2E2(V1y_2E0,V0x_2E0),c_2Ereal_2E_2F_2E2(V2z_2E0,V0x_2E0)) = c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__add_2E2(V1y_2E0,V2z_2E0),V0x_2E0) ) ).
tff(thm_2Ereal_2EREAL__ADD__RAT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( ( V1b_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V3d_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2E_2F_2E2(V0a_2E0,V1b_2E0),c_2Ereal_2E_2F_2E2(V2c_2E0,V3d_2E0)) = c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3d_2E0),c_2Erealax_2Ereal__mul_2E2(V1b_2E0,V2c_2E0)),c_2Erealax_2Ereal__mul_2E2(V1b_2E0,V3d_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__SUB__RAT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( ( V1b_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V3d_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2E_2F_2E2(V0a_2E0,V1b_2E0),c_2Ereal_2E_2F_2E2(V2c_2E0,V3d_2E0)) = c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3d_2E0),c_2Erealax_2Ereal__mul_2E2(V1b_2E0,V2c_2E0)),c_2Erealax_2Ereal__mul_2E2(V1b_2E0,V3d_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__SUB,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0m_2E0),V1n_2E0))
<=> ( c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0) = c_2Enum_2E0_2E0 ) )
=> ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(V0m_2E0),c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0m_2E0),V1n_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2D_2E2(V1n_2E0,V0m_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2D_2E2(V0m_2E0,V1n_2E0))) ) ) ).
tff(thm_2Ereal_2EREAL__POS__POS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Epos_2E1(V0x_2E0))) ).
tff(thm_2Ereal_2EREAL__POS__ID,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Ereal_2Epos_2E1(V0x_2E0) = V0x_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__POS__INFLATE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Epos_2E1(V0x_2E0))) ).
tff(thm_2Ereal_2EREAL__POS__MONO,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Epos_2E1(V0x_2E0),c_2Ereal_2Epos_2E1(V1y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__POS__EQ__ZERO,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epos_2E1(V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Ereal_2EREAL__POS__LE__ZERO,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Epos_2E1(V0x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ).
tff(thm_2Ereal_2EREAL__MIN__REFL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emin_2E2(V0x_2E0,V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__LE__MIN,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0z_2E0,c_2Ereal_2Emin_2E2(V1x_2E0,V2y_2E0)))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V0z_2E0,V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0z_2E0,V2y_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__MIN__LE,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Emin_2E2(V1x_2E0,V2y_2E0),V0z_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V0z_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V2y_2E0,V0z_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__MIN__LE1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0),V0x_2E0)) ).
tff(thm_2Ereal_2EREAL__MIN__LE2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0),V1y_2E0)) ).
tff(thm_2Ereal_2EREAL__LT__MIN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0)))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,V1y_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__MIN__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0),V2z_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V2z_2E0))
| p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V2z_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__MIN__ALT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
=> ( c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0) = V0x_2E0 ) )
& ( p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V0x_2E0))
=> ( c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0) = V1y_2E0 ) ) ) ).
tff(thm_2Ereal_2EREAL__MIN__LE__LIN,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0z_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0z_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Emin_2E2(V1x_2E0,V2y_2E0),c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0z_2E0,V1x_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0z_2E0),V2y_2E0)))) ) ).
tff(thm_2Ereal_2EREAL__MIN__ADD,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emin_2E2(c_2Erealax_2Ereal__add_2E2(V1x_2E0,V0z_2E0),c_2Erealax_2Ereal__add_2E2(V2y_2E0,V0z_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Emin_2E2(V1x_2E0,V2y_2E0),V0z_2E0) ) ).
tff(thm_2Ereal_2EREAL__MIN__SUB,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emin_2E2(c_2Ereal_2Ereal__sub_2E2(V1x_2E0,V0z_2E0),c_2Ereal_2Ereal__sub_2E2(V2y_2E0,V0z_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Emin_2E2(V1x_2E0,V2y_2E0),V0z_2E0) ) ).
tff(thm_2Ereal_2EREAL__IMP__MIN__LE2,axiom,
! [V0x1_2E0: tyop_2Erealax_2Ereal,V1x2_2E0: tyop_2Erealax_2Ereal,V2y1_2E0: tyop_2Erealax_2Ereal,V3y2_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0x1_2E0,V2y1_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x2_2E0,V3y2_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Emin_2E2(V0x1_2E0,V1x2_2E0),c_2Ereal_2Emin_2E2(V2y1_2E0,V3y2_2E0))) ) ).
tff(thm_2Ereal_2EREAL__MIN__ACI,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0) = c_2Ereal_2Emin_2E2(V1y_2E0,V0x_2E0) )
& ( c_2Ereal_2Emin_2E2(c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0),V2z_2E0) = c_2Ereal_2Emin_2E2(V0x_2E0,c_2Ereal_2Emin_2E2(V1y_2E0,V2z_2E0)) )
& ( c_2Ereal_2Emin_2E2(V0x_2E0,c_2Ereal_2Emin_2E2(V1y_2E0,V2z_2E0)) = c_2Ereal_2Emin_2E2(V1y_2E0,c_2Ereal_2Emin_2E2(V0x_2E0,V2z_2E0)) )
& ( c_2Ereal_2Emin_2E2(V0x_2E0,V0x_2E0) = V0x_2E0 )
& ( c_2Ereal_2Emin_2E2(V0x_2E0,c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__MAX__REFL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emax_2E2(V0x_2E0,V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Ereal_2EREAL__LE__MAX,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0z_2E0,c_2Ereal_2Emax_2E2(V1x_2E0,V2y_2E0)))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V0z_2E0,V1x_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V0z_2E0,V2y_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__LE__MAX1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Emax_2E2(V0x_2E0,V1y_2E0))) ).
tff(thm_2Ereal_2EREAL__LE__MAX2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,c_2Ereal_2Emax_2E2(V0x_2E0,V1y_2E0))) ).
tff(thm_2Ereal_2EREAL__MAX__LE,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Emax_2E2(V1x_2E0,V2y_2E0),V0z_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V0z_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2y_2E0,V0z_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__LT__MAX,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,c_2Ereal_2Emax_2E2(V0x_2E0,V1y_2E0)))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,V0x_2E0))
| p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,V1y_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__MAX__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Emax_2E2(V0x_2E0,V1y_2E0),V2z_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V2z_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V2z_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__MAX__ALT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
=> ( c_2Ereal_2Emax_2E2(V0x_2E0,V1y_2E0) = V1y_2E0 ) )
& ( p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V0x_2E0))
=> ( c_2Ereal_2Emax_2E2(V0x_2E0,V1y_2E0) = V0x_2E0 ) ) ) ).
tff(thm_2Ereal_2EREAL__MAX__MIN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emax_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Emin_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),c_2Erealax_2Ereal__neg_2E1(V1y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__MIN__MAX,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Emax_2E2(c_2Erealax_2Ereal__neg_2E1(V0x_2E0),c_2Erealax_2Ereal__neg_2E1(V1y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__LIN__LE__MAX,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0z_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0z_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0z_2E0,V1x_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0z_2E0),V2y_2E0)),c_2Ereal_2Emax_2E2(V1x_2E0,V2y_2E0))) ) ).
tff(thm_2Ereal_2EREAL__MAX__ADD,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emax_2E2(c_2Erealax_2Ereal__add_2E2(V1x_2E0,V0z_2E0),c_2Erealax_2Ereal__add_2E2(V2y_2E0,V0z_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Emax_2E2(V1x_2E0,V2y_2E0),V0z_2E0) ) ).
tff(thm_2Ereal_2EREAL__MAX__SUB,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emax_2E2(c_2Ereal_2Ereal__sub_2E2(V1x_2E0,V0z_2E0),c_2Ereal_2Ereal__sub_2E2(V2y_2E0,V0z_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Emax_2E2(V1x_2E0,V2y_2E0),V0z_2E0) ) ).
tff(thm_2Ereal_2EREAL__IMP__MAX__LE2,axiom,
! [V0x1_2E0: tyop_2Erealax_2Ereal,V1x2_2E0: tyop_2Erealax_2Ereal,V2y1_2E0: tyop_2Erealax_2Ereal,V3y2_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0x1_2E0,V2y1_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x2_2E0,V3y2_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Emax_2E2(V0x1_2E0,V1x2_2E0),c_2Ereal_2Emax_2E2(V2y1_2E0,V3y2_2E0))) ) ).
tff(thm_2Ereal_2EREAL__MAX__ACI,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Emax_2E2(V0x_2E0,V1y_2E0) = c_2Ereal_2Emax_2E2(V1y_2E0,V0x_2E0) )
& ( c_2Ereal_2Emax_2E2(c_2Ereal_2Emax_2E2(V0x_2E0,V1y_2E0),V2z_2E0) = c_2Ereal_2Emax_2E2(V0x_2E0,c_2Ereal_2Emax_2E2(V1y_2E0,V2z_2E0)) )
& ( c_2Ereal_2Emax_2E2(V0x_2E0,c_2Ereal_2Emax_2E2(V1y_2E0,V2z_2E0)) = c_2Ereal_2Emax_2E2(V1y_2E0,c_2Ereal_2Emax_2E2(V0x_2E0,V2z_2E0)) )
& ( c_2Ereal_2Emax_2E2(V0x_2E0,V0x_2E0) = V0x_2E0 )
& ( c_2Ereal_2Emax_2E2(V0x_2E0,c_2Ereal_2Emax_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2Emax_2E2(V0x_2E0,V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__SUP__EXISTS__UNIQUE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4s_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0p_2E0),V4s_2E0))
<=> ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( ? [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V6x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V5y_2E0,V6x_2E0)) )
<=> p(c_2Erealax_2Ereal__lt_2E2(V5y_2E0,V4s_2E0)) ) )
=> ! [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))
& ? [V2z_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V3x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2z_2E0)) ) )
=> p(c_2Ebool_2E_3F_21_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0p_2E0))) ) ) ).
tff(thm_2Ereal_2EREAL__SUP__MAX,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1z_2E0: tyop_2Erealax_2Ereal] :
( ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V1z_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V2x_2E0,V1z_2E0)) ) )
=> ( c_2Ereal_2Esup_2E1(V0p_2E0) = V1z_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__IMP__SUP__LE,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( ? [V2r_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V2r_2E0))
& ! [V3r_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V3r_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V3r_2E0,V1x_2E0)) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esup_2E1(V0p_2E0),V1x_2E0)) ) ).
tff(thm_2Ereal_2EREAL__IMP__LE__SUP,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( ? [V2r_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V2r_2E0))
& ? [V3z_2E0: tyop_2Erealax_2Ereal] :
! [V4r_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V4r_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V4r_2E0,V3z_2E0)) )
& ? [V5r_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V5r_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V5r_2E0)) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,c_2Ereal_2Esup_2E1(V0p_2E0))) ) ).
tff(thm_2Ereal_2EREAL__INF__MIN,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1z_2E0: tyop_2Erealax_2Ereal] :
( ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V1z_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V1z_2E0,V2x_2E0)) ) )
=> ( c_2Ereal_2Einf_2E1(V0p_2E0) = V1z_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__IMP__LE__INF,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1r_2E0: tyop_2Erealax_2Ereal] :
( ( ? [V2x_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V3x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V1r_2E0,V3x_2E0)) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(V1r_2E0,c_2Ereal_2Einf_2E1(V0p_2E0))) ) ).
tff(thm_2Ereal_2EREAL__IMP__INF__LE,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1r_2E0: tyop_2Erealax_2Ereal] :
( ( ? [V2z_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V3x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V2z_2E0,V3x_2E0)) )
& ? [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V4x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V4x_2E0,V1r_2E0)) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Einf_2E1(V0p_2E0),V1r_2E0)) ) ).
tff(thm_2Ereal_2EREAL__INF__LT,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1z_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_2Ereal_2Einf_2E1(V0p_2E0),V1z_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(V3x_2E0,V1z_2E0)) ) ) ).
tff(thm_2Ereal_2EREAL__INF__CLOSE,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1e_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_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_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(V3x_2E0,c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Einf_2E1(V0p_2E0),V1e_2E0))) ) ) ).
tff(thm_2Ereal_2ESUP__EPSILON,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_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_2Ereal_2Ereal__lte_2E2(V4x_2E0,V3z_2E0)) ) )
=> ? [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V5x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esup_2E1(V0p_2E0),c_2Erealax_2Ereal__add_2E2(V5x_2E0,V1e_2E0))) ) ) ).
tff(thm_2Ereal_2EREAL__LE__SUP,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( ? [V2y_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V2y_2E0))
& ? [V3y_2E0: tyop_2Erealax_2Ereal] :
! [V4z_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V4z_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V4z_2E0,V3y_2E0)) ) )
=> ( p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,c_2Ereal_2Esup_2E1(V0p_2E0)))
<=> ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( ! [V6z_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V6z_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V6z_2E0,V5y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V5y_2E0)) ) ) ) ).
tff(thm_2Ereal_2EREAL__INF__LE,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( ? [V2y_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V2y_2E0))
& ? [V3y_2E0: tyop_2Erealax_2Ereal] :
! [V4z_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V4z_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V3y_2E0,V4z_2E0)) ) )
=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Einf_2E1(V0p_2E0),V1x_2E0))
<=> ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( ! [V6z_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V6z_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V5y_2E0,V6z_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V5y_2E0,V1x_2E0)) ) ) ) ).
tff(thm_2Ereal_2EREAL__SUP__CONST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1r_2E0))
<=> ( V1r_2E0 = V0x_2E0 ) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Esup_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0)) = V0x_2E0 ) ) ).
tff(thm_2Ereal_2EREAL__MUL__SUB2__CANCEL,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V1x_2E0,V2y_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(V0z_2E0,V1x_2E0),V2y_2E0)) = c_2Erealax_2Ereal__mul_2E2(V0z_2E0,V2y_2E0) ) ).
tff(thm_2Ereal_2EREAL__MUL__SUB1__CANCEL,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V2y_2E0,V1x_2E0),c_2Erealax_2Ereal__mul_2E2(V2y_2E0,c_2Ereal_2Ereal__sub_2E2(V0z_2E0,V1x_2E0))) = c_2Erealax_2Ereal__mul_2E2(V2y_2E0,V0z_2E0) ) ).
tff(thm_2Ereal_2EREAL__NEG__HALF,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(V0x_2E0,c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ).
tff(thm_2Ereal_2EREAL__NEG__THIRD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(V0x_2E0,c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) = c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ereal_2EREAL__DIV__DENOM__CANCEL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal_2E_2F_2E2(c_2Ereal_2E_2F_2E2(V1y_2E0,V0x_2E0),c_2Ereal_2E_2F_2E2(V2z_2E0,V0x_2E0)) = c_2Ereal_2E_2F_2E2(V1y_2E0,V2z_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__DIV__DENOM__CANCEL2,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(c_2Ereal_2E_2F_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2E_2F_2E2(V1z_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Ereal_2E_2F_2E2(V0y_2E0,V1z_2E0) ) ).
tff(thm_2Ereal_2EREAL__DIV__DENOM__CANCEL3,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(c_2Ereal_2E_2F_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Ereal_2E_2F_2E2(V1z_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) = c_2Ereal_2E_2F_2E2(V0y_2E0,V1z_2E0) ) ).
tff(thm_2Ereal_2EREAL__DIV__INNER__CANCEL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(V1y_2E0,V0x_2E0),c_2Ereal_2E_2F_2E2(V0x_2E0,V2z_2E0)) = c_2Ereal_2E_2F_2E2(V1y_2E0,V2z_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__DIV__INNER__CANCEL2,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V1z_2E0)) = c_2Ereal_2E_2F_2E2(V0y_2E0,V1z_2E0) ) ).
tff(thm_2Ereal_2EREAL__DIV__INNER__CANCEL3,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V1z_2E0)) = c_2Ereal_2E_2F_2E2(V0y_2E0,V1z_2E0) ) ).
tff(thm_2Ereal_2EREAL__DIV__OUTER__CANCEL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,V1y_2E0),c_2Ereal_2E_2F_2E2(V2z_2E0,V0x_2E0)) = c_2Ereal_2E_2F_2E2(V2z_2E0,V1y_2E0) ) ) ).
tff(thm_2Ereal_2EREAL__DIV__OUTER__CANCEL2,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0y_2E0),c_2Ereal_2E_2F_2E2(V1z_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Ereal_2E_2F_2E2(V1z_2E0,V0y_2E0) ) ).
tff(thm_2Ereal_2EREAL__DIV__OUTER__CANCEL3,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0y_2E0),c_2Ereal_2E_2F_2E2(V1z_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) = c_2Ereal_2E_2F_2E2(V1z_2E0,V0y_2E0) ) ).
tff(thm_2Ereal_2EREAL__DIV__REFL2,axiom,
c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Ereal_2EREAL__DIV__REFL3,axiom,
c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Ereal_2EREAL__HALF__BETWEEN,axiom,
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ereal_2EREAL__THIRDS__BETWEEN,axiom,
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ereal_2EREAL__LE__SUB__CANCEL2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V2z_2E0),c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V2z_2E0)) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ereal_2EREAL__ADD__SUB__ALT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),V1y_2E0) = V0x_2E0 ) ).
tff(thm_2Ereal_2EINFINITE__REAL__UNIV,axiom,
~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal_2Eadd__rat,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0y_2E0))
<=> ( V0y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( ! [V0y_2E0: tyop_2Erealax_2Ereal,V2v_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0y_2E0),V2v_2E0))
<=> ( V0y_2E0 = V2v_2E0 ) )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2v_2E0: tyop_2Erealax_2Ereal,V3u_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0),c_2Ereal_2E_2F_2E2(V3u_2E0,V2v_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0y_2E0),c_2Erealax_2Ereal__add_2E2(c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0)),c_2Ereal_2E_2F_2E2(V3u_2E0,V2v_2E0)),c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V2v_2E0),c_2Erealax_2Ereal__add_2E2(c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0),c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V3u_2E0,V2v_2E0))),c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0y_2E0),V2v_2E0),c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__add_2E2(V1x_2E0,V3u_2E0),V2v_2E0),c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V1x_2E0,V2v_2E0),c_2Erealax_2Ereal__mul_2E2(V3u_2E0,V0y_2E0)),c_2Erealax_2Ereal__mul_2E2(V0y_2E0,V2v_2E0))))) ) ) ) ).
tff(thm_2Ereal_2Eadd__ratl,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V1y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V1y_2E0))
<=> ( V1y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ! [V0z_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2E_2F_2E2(V2x_2E0,V1y_2E0),V0z_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V1y_2E0),c_2Erealax_2Ereal__add_2E2(c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V2x_2E0,V1y_2E0)),V0z_2E0),c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__add_2E2(V2x_2E0,c_2Erealax_2Ereal__mul_2E2(V0z_2E0,V1y_2E0)),V1y_2E0)) ) ) ).
tff(thm_2Ereal_2Eadd__ratr,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V0z_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0z_2E0))
<=> ( V0z_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ! [V0z_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(V2x_2E0,c_2Ereal_2E_2F_2E2(V1y_2E0,V0z_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0z_2E0),c_2Erealax_2Ereal__add_2E2(V2x_2E0,c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1y_2E0,V0z_2E0))),c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V2x_2E0,V0z_2E0),V1y_2E0),V0z_2E0)) ) ) ).
tff(thm_2Ereal_2Eadd__ints,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Ereal_2Ereal__of__num_2E1(V1m_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0)) )
& ( c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)),c_2Ereal_2Ereal__of__num_2E1(V1m_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,V1m_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2D_2E2(V1m_2E0,V0n_2E0)),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2D_2E2(V0n_2E0,V1m_2E0)))) )
& ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V1m_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Eprim__rec_2E_3C_2E2(V0n_2E0,V1m_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2D_2E2(V1m_2E0,V0n_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2D_2E2(V0n_2E0,V1m_2E0))) )
& ( c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V1m_2E0))) = c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2B_2E2(V0n_2E0,V1m_2E0))) ) ) ).
tff(thm_2Ereal_2Emult__rat,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0y_2E0))
<=> ( V0y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2v_2E0: tyop_2Erealax_2Ereal,V3u_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0),c_2Ereal_2E_2F_2E2(V3u_2E0,V2v_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0y_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0)),c_2Ereal_2E_2F_2E2(V3u_2E0,V2v_2E0)),c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V2v_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0),c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V3u_2E0,V2v_2E0))),c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__mul_2E2(V1x_2E0,V3u_2E0),c_2Erealax_2Ereal__mul_2E2(V0y_2E0,V2v_2E0)))) ) ) ).
tff(thm_2Ereal_2Emult__ratl,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V1y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V1y_2E0))
<=> ( V1y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ! [V0z_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(V2x_2E0,V1y_2E0),V0z_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V1y_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V2x_2E0,V1y_2E0)),V0z_2E0),c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__mul_2E2(V2x_2E0,V0z_2E0),V1y_2E0)) ) ) ).
tff(thm_2Ereal_2Emult__ratr,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V0z_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0z_2E0))
<=> ( V0z_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ! [V0z_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(V2x_2E0,c_2Ereal_2E_2F_2E2(V1y_2E0,V0z_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0z_2E0),c_2Erealax_2Ereal__mul_2E2(V2x_2E0,c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1y_2E0,V0z_2E0))),c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__mul_2E2(V2x_2E0,V1y_2E0),V0z_2E0)) ) ) ).
tff(thm_2Ereal_2Emult__ints,axiom,
! [V0b_2E0: tyop_2Enum_2Enum,V1a_2E0: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1a_2E0),c_2Ereal_2Ereal__of__num_2E1(V0b_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2A_2E2(V1a_2E0,V0b_2E0)) )
& ( c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V1a_2E0)),c_2Ereal_2Ereal__of__num_2E1(V0b_2E0)) = c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2A_2E2(V1a_2E0,V0b_2E0))) )
& ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1a_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V0b_2E0))) = c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2A_2E2(V1a_2E0,V0b_2E0))) )
& ( c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V1a_2E0)),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V0b_2E0))) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2A_2E2(V1a_2E0,V0b_2E0)) ) ) ).
tff(thm_2Ereal_2Epow__rat,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V4x_2E0) = V4x_2E0 )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum,V3a_2E0: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Epow_2E2(V1x_2E0,c_2Enum_2E0_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
& ( c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V2n_2E0))) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V2n_2E0))) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(V3a_2E0)),c_2Earithmetic_2ENUMERAL_2E1(V2n_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(V3a_2E0),c_2Earithmetic_2ENUMERAL_2E1(V2n_2E0))) )
& ( c_2Ereal_2Epow_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(V3a_2E0))),c_2Earithmetic_2ENUMERAL_2E1(V2n_2E0)) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Earithmetic_2EODD_2E1(c_2Earithmetic_2ENUMERAL_2E1(V2n_2E0)),c_2Erealax_2Ereal__neg_2E0,F0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EEXP_2E2(c_2Earithmetic_2ENUMERAL_2E1(V3a_2E0),c_2Earithmetic_2ENUMERAL_2E1(V2n_2E0)))) )
& ( c_2Ereal_2Epow_2E2(c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0),V2n_2E0) = c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_2E2(V1x_2E0,V2n_2E0),c_2Ereal_2Epow_2E2(V0y_2E0,V2n_2E0)) ) ) ) ).
tff(thm_2Ereal_2Eneg__rat,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0y_2E0))
<=> ( V0y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( ! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0y_2E0))
<=> ( V0y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0y_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0))),c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__neg_2E1(V1x_2E0),V0y_2E0)) )
& ( c_2Ereal_2E_2F_2E2(V1x_2E0,c_2Erealax_2Ereal__neg_2E1(V0y_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0y_2E0),c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0)),c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__neg_2E1(V1x_2E0),V0y_2E0)) ) ) ) ) ).
tff(thm_2Ereal_2Eeq__rat,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0y_2E0))
<=> ( V0y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( ! [V0y_2E0: tyop_2Erealax_2Ereal,V2v_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F3_2E0,V0y_2E0),V2v_2E0))
<=> ( V0y_2E0 = V2v_2E0 ) )
=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal,V0y_2E0: tyop_2Erealax_2Ereal,V3u_2E0: tyop_2Erealax_2Ereal,V2v_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0,V1x_2E0),V0y_2E0),V3u_2E0),V2v_2E0))
<=> ( c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0) = c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V3u_2E0,V2v_2E0)) ) )
=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal,V0y_2E0: tyop_2Erealax_2Ereal,V3u_2E0: tyop_2Erealax_2Ereal,V2v_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V1x_2E0),V0y_2E0),V3u_2E0),V2v_2E0))
<=> ( c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0)) = c_2Ereal_2E_2F_2E2(V3u_2E0,V2v_2E0) ) )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2v_2E0: tyop_2Erealax_2Ereal,V3u_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0) = c_2Ereal_2E_2F_2E2(V3u_2E0,V2v_2E0) )
<=> p(c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V1x_2E0),V0y_2E0),V3u_2E0),V2v_2E0),c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V2v_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0,V1x_2E0),V0y_2E0),V3u_2E0),V2v_2E0),c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F3_2E0,V0y_2E0),V2v_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F3_2E0,V1x_2E0),V3u_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F3_2E0,c_2Erealax_2Ereal__mul_2E2(V1x_2E0,V2v_2E0)),c_2Erealax_2Ereal__mul_2E2(V0y_2E0,V3u_2E0)))))) ) ) ) ) ) ).
tff(thm_2Ereal_2Eeq__ratl,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V1y_2E0))
<=> ( V1y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( ! [V2x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V0z_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V2x_2E0),V1y_2E0),V0z_2E0))
<=> ( c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V2x_2E0,V1y_2E0)) = V0z_2E0 ) )
=> ( ! [V2x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V0z_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V2x_2E0),V1y_2E0),V0z_2E0))
<=> ( V2x_2E0 = c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V0z_2E0) ) )
=> ! [V0z_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2E_2F_2E2(V2x_2E0,V1y_2E0) = V0z_2E0 )
<=> p(c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V1y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V2x_2E0),V1y_2E0),V0z_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V2x_2E0),V1y_2E0),V0z_2E0))) ) ) ) ) ).
tff(thm_2Ereal_2Eeq__ratr,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V1y_2E0))
<=> ( V1y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( ! [V1y_2E0: tyop_2Erealax_2Ereal,V0z_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V1y_2E0),V0z_2E0),V2x_2E0))
<=> ( c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V0z_2E0) = V2x_2E0 ) )
=> ( ! [V0z_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0z_2E0),V2x_2E0),V1y_2E0))
<=> ( V0z_2E0 = c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V2x_2E0,V1y_2E0)) ) )
=> ! [V0z_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( ( V0z_2E0 = c_2Ereal_2E_2F_2E2(V2x_2E0,V1y_2E0) )
<=> p(c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V1y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0z_2E0),V2x_2E0),V1y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V1y_2E0),V0z_2E0),V2x_2E0))) ) ) ) ) ).
tff(thm_2Ereal_2Eeq__ints,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( ( c_2Ereal_2Ereal__of__num_2E1(V0n_2E0) = c_2Ereal_2Ereal__of__num_2E1(V1m_2E0) )
<=> ( V0n_2E0 = V1m_2E0 ) )
& ( ( c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)) = c_2Ereal_2Ereal__of__num_2E1(V1m_2E0) )
<=> ( ( V0n_2E0 = c_2Enum_2E0_2E0 )
& ( V1m_2E0 = c_2Enum_2E0_2E0 ) ) )
& ( ( c_2Ereal_2Ereal__of__num_2E1(V0n_2E0) = c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V1m_2E0)) )
<=> ( ( V0n_2E0 = c_2Enum_2E0_2E0 )
& ( V1m_2E0 = c_2Enum_2E0_2E0 ) ) )
& ( ( c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)) = c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V1m_2E0)) )
<=> ( V0n_2E0 = V1m_2E0 ) ) ) ).
tff(thm_2Ereal_2Ediv__ratr,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V1y_2E0: tyop_2Erealax_2Ereal,V0z_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V1y_2E0),V0z_2E0))
<=> ( ( V1y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V0z_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
=> ! [V0z_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(V2x_2E0,c_2Ereal_2E_2F_2E2(V1y_2E0,V0z_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V1y_2E0),V0z_2E0),c_2Ereal_2E_2F_2E2(V2x_2E0,c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1y_2E0,V0z_2E0))),c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__mul_2E2(V2x_2E0,V0z_2E0),V1y_2E0)) ) ) ).
tff(thm_2Ereal_2Ediv__ratl,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V1y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V1y_2E0))
<=> ( V1y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ! [V0z_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(c_2Ereal_2E_2F_2E2(V2x_2E0,V1y_2E0),V0z_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V1y_2E0),c_2Ereal_2E_2F_2E2(c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V2x_2E0,V1y_2E0)),V0z_2E0),c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0z_2E0),c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(c_2Ereal_2E_2F_2E2(V2x_2E0,V1y_2E0),V0z_2E0)),c_2Ereal_2E_2F_2E2(V2x_2E0,c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V0z_2E0)))) ) ) ).
tff(thm_2Ereal_2Ediv__rat,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0y_2E0))
<=> ( V0y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( ! [V3u_2E0: tyop_2Erealax_2Ereal,V2v_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V3u_2E0),V2v_2E0))
<=> ( ( V3u_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V2v_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2v_2E0: tyop_2Erealax_2Ereal,V3u_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0),c_2Ereal_2E_2F_2E2(V3u_2E0,V2v_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V3u_2E0),V2v_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0),c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V3u_2E0,V2v_2E0))),c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0y_2E0),c_2Ereal_2E_2F_2E2(c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0)),c_2Ereal_2E_2F_2E2(V3u_2E0,V2v_2E0)),c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__mul_2E2(V1x_2E0,V2v_2E0),c_2Erealax_2Ereal__mul_2E2(V0y_2E0,V3u_2E0)))) ) ) ) ).
tff(thm_2Ereal_2Ele__rat,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2n_2E0))
<=> ( V2n_2E0 = c_2Enum_2E0_2E0 ) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1u_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(V2n_2E0)),c_2Ereal_2E_2F_2E2(V1u_2E0,c_2Ereal_2Ereal__of__num_2E1(V3m_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2n_2E0),c_2Ereal_2Ereal__lte_2E2(c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))),c_2Ereal_2E_2F_2E2(V1u_2E0,c_2Ereal_2Ereal__of__num_2E1(V3m_2E0))),c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V3m_2E0),c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(V2n_2E0)),c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1u_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))),c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V3m_2E0),V0x_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V2n_2E0),V1u_2E0)))) ) ) ).
tff(thm_2Ereal_2Ele__ratl,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2n_2E0))
<=> ( V2n_2E0 = c_2Enum_2E0_2E0 ) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1u_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(V2n_2E0)),V1u_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2n_2E0),c_2Ereal_2Ereal__lte_2E2(c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))),V1u_2E0),c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V2n_2E0),V1u_2E0))) ) ) ).
tff(thm_2Ereal_2Ele__ratr,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V2m_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2m_2E0))
<=> ( V2m_2E0 = c_2Enum_2E0_2E0 ) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1u_2E0: tyop_2Erealax_2Ereal,V2m_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2E_2F_2E2(V1u_2E0,c_2Ereal_2Ereal__of__num_2E1(V2m_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2m_2E0),c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1u_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))),c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V2m_2E0),V0x_2E0),V1u_2E0)) ) ) ).
tff(thm_2Ereal_2Ele__int,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Ereal_2Ereal__of__num_2E1(V1m_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,V1m_2E0) )
& ( c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)),c_2Ereal_2Ereal__of__num_2E1(V1m_2E0)) = c_2Ebool_2ET_2E0 )
& ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V1m_2E0))))
<=> ( ( V0n_2E0 = c_2Enum_2E0_2E0 )
& ( V1m_2E0 = c_2Enum_2E0_2E0 ) ) )
& ( c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V1m_2E0))) = c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V0n_2E0) ) ) ).
tff(thm_2Ereal_2Elt__rat,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2n_2E0))
<=> ( V2n_2E0 = c_2Enum_2E0_2E0 ) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1u_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum,V3m_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(V2n_2E0)),c_2Ereal_2E_2F_2E2(V1u_2E0,c_2Ereal_2Ereal__of__num_2E1(V3m_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2n_2E0),c_2Erealax_2Ereal__lt_2E2(c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))),c_2Ereal_2E_2F_2E2(V1u_2E0,c_2Ereal_2Ereal__of__num_2E1(V3m_2E0))),c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V3m_2E0),c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(V2n_2E0)),c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1u_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))),c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V3m_2E0),V0x_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V2n_2E0),V1u_2E0)))) ) ) ).
tff(thm_2Ereal_2Elt__ratl,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2n_2E0))
<=> ( V2n_2E0 = c_2Enum_2E0_2E0 ) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1u_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(V2n_2E0)),V1u_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2n_2E0),c_2Erealax_2Ereal__lt_2E2(c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))),V1u_2E0),c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V2n_2E0),V1u_2E0))) ) ) ).
tff(thm_2Ereal_2Elt__ratr,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V2m_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2m_2E0))
<=> ( V2m_2E0 = c_2Enum_2E0_2E0 ) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1u_2E0: tyop_2Erealax_2Ereal,V2m_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2E_2F_2E2(V1u_2E0,c_2Ereal_2Ereal__of__num_2E1(V2m_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2m_2E0),c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Emarker_2Eunint_2E1(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(V1u_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))),c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V2m_2E0),V0x_2E0),V1u_2E0)) ) ) ).
tff(thm_2Ereal_2Elt__int,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Ereal_2Ereal__of__num_2E1(V1m_2E0)) = c_2Eprim__rec_2E_3C_2E2(V0n_2E0,V1m_2E0) )
& ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)),c_2Ereal_2Ereal__of__num_2E1(V1m_2E0)))
<=> ( ( V0n_2E0 != c_2Enum_2E0_2E0 )
| ( V1m_2E0 != c_2Enum_2E0_2E0 ) ) )
& ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V1m_2E0))) = c_2Ebool_2EF_2E0 )
& ( c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V1m_2E0))) = c_2Eprim__rec_2E_3C_2E2(V1m_2E0,V0n_2E0) ) ) ).
tff(thm_2Ereal_2ENUM__FLOOR__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Ereal_2ENUM__FLOOR_2E1(V0x_2E0)),V0x_2E0)) ) ).
tff(thm_2Ereal_2ENUM__FLOOR__LE2,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0y_2E0))
=> ( c_2Earithmetic_2E_3C_3D_2E2(V1x_2E0,c_2Ereal_2ENUM__FLOOR_2E1(V0y_2E0)) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(V1x_2E0),V0y_2E0) ) ) ).
tff(thm_2Ereal_2ENUM__FLOOR__LET,axiom,
! [V0y_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Earithmetic_2E_3C_3D_2E2(c_2Ereal_2ENUM__FLOOR_2E1(V1x_2E0),V0y_2E0) = c_2Erealax_2Ereal__lt_2E2(V1x_2E0,c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(V0y_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ereal_2ENUM__FLOOR__DIV,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Ereal_2ENUM__FLOOR_2E1(c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0))),V0y_2E0),V1x_2E0)) ) ).
tff(thm_2Ereal_2ENUM__FLOOR__DIV__LOWERBOUND,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(V1x_2E0,c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2B_2E2(c_2Ereal_2ENUM__FLOOR_2E1(c_2Ereal_2E_2F_2E2(V1x_2E0,V0y_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0y_2E0))) ) ).
tff(thm_2Ereal_2ENUM__FLOOR__BASE,axiom,
! [V0r_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V0r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
=> ( c_2Ereal_2ENUM__FLOOR_2E1(V0r_2E0) = c_2Enum_2E0_2E0 ) ) ).
tff(thm_2Ereal_2ENUM__FLOOR__EQNS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] :
( ( c_2Ereal_2ENUM__FLOOR_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)) = V0n_2E0 )
& ( c_2Ereal_2ENUM__FLOOR_2E1(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0))) = c_2Enum_2E0_2E0 )
& ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1m_2E0))
=> ( c_2Ereal_2ENUM__FLOOR_2E1(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Ereal_2Ereal__of__num_2E1(V1m_2E0))) = c_2Earithmetic_2EDIV_2E2(V0n_2E0,V1m_2E0) ) )
& ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1m_2E0))
=> ( c_2Ereal_2ENUM__FLOOR_2E1(c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)),c_2Ereal_2Ereal__of__num_2E1(V1m_2E0))) = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Ereal_2ENUM__FLOOR__LOWER__BOUND,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(c_2Ereal_2ENUM__FLOOR_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),V1n_2E0) ) ).
tff(thm_2Ereal_2ENUM__FLOOR__upper__bound,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0),V0x_2E0) = c_2Eprim__rec_2E_3C_2E2(V1n_2E0,c_2Ereal_2ENUM__FLOOR_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) ) ).
tff(thm_2Ereal_2ENUM__CEILING__NUM__FLOOR,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] :
( ! [V0r_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F1_2E0,V0r_2E0),V1n_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V0r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| ( V0r_2E0 = c_2Ereal_2Ereal__of__num_2E1(V1n_2E0) ) ) )
=> ( ! [V0r_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F0_2E0,V0r_2E0),V1n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F1_2E0,V0r_2E0),V1n_2E0),V1n_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) )
=> ! [V0r_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2ENUM__CEILING_2E1(V0r_2E0) = c_2Ebool_2ELET_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F0_2E0,V0r_2E0),c_2Ereal_2ENUM__FLOOR_2E1(V0r_2E0)) ) ) ) ).
tff(thm_2Ereal_2ELE__NUM__CEILING,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Ereal_2ENUM__CEILING_2E1(V0x_2E0)))) ).
tff(thm_2Ereal_2ENUM__CEILING__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)))
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Ereal_2ENUM__CEILING_2E1(V0x_2E0),V1n_2E0)) ) ).
%------------------------------------------------------------------------------