ITP001 Axioms: ITP143_7.ax
%------------------------------------------------------------------------------
% File : ITP143_7 : TPTP v9.0.0. 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 : extreal.ax [Gau19]
% : HL4143_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 731 ( 202 unt; 217 typ; 0 def)
% Number of atoms : 1337 ( 702 equ)
% Maximal formula atoms : 17 ( 1 avg)
% Number of connectives : 1028 ( 205 ~; 30 |; 351 &)
% ( 80 <=>; 362 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 5 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 158 ( 101 >; 57 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 212 ( 212 usr; 70 con; 0-5 aty)
% Number of variables : 1552 (1412 !; 30 ?;1552 :)
% ( 110 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Eextreal_2Eextreal,type,
tyop_2Eextreal_2Eextreal: $tType ).
tff(tyop_2Eind__type_2Erecspace,type,
tyop_2Eind__type_2Erecspace: $tType > $tType ).
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_2Eseq_2E_2D_2D_3E_2E0,type,
c_2Eseq_2E_2D_2D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Eseq_2E_2D_2D_3E_2E2,type,
c_2Eseq_2E_2D_2D_3E_2E2: ( tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
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_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_2Ebool_2E_3F_2E0,type,
c_2Ebool_2E_3F_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_2E1,type,
c_2Ebool_2E_3F_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_40_2E0,type,
c_2Emin_2E_40_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a) ).
tff(c_2Emin_2E_40_2E1,type,
c_2Emin_2E_40_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > A_27a ) ).
tff(c_2Ebool_2EARB_2E0,type,
c_2Ebool_2EARB_2E0:
!>[A_27a: $tType] : 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_2Eind__type_2EBOTTOM_2E0,type,
c_2Eind__type_2EBOTTOM_2E0:
!>[A_27a: $tType] : tyop_2Eind__type_2Erecspace(A_27a) ).
tff(c_2Epred__set_2ECARD_2E0,type,
c_2Epred__set_2ECARD_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum) ).
tff(c_2Epred__set_2ECARD_2E1,type,
c_2Epred__set_2ECARD_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > 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_2Eind__type_2ECONSTR_2E0,type,
c_2Eind__type_2ECONSTR_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(A_27a)),tyop_2Eind__type_2Erecspace(A_27a)))) ).
tff(c_2Eind__type_2ECONSTR_2E3,type,
c_2Eind__type_2ECONSTR_2E3:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum * A_27a * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(A_27a)) ) > tyop_2Eind__type_2Erecspace(A_27a) ) ).
tff(c_2Epred__set_2ECROSS_2E0,type,
c_2Epred__set_2ECROSS_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2ECROSS_2E2,type,
c_2Epred__set_2ECROSS_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool) ) ).
tff(c_2Ebool_2EDATATYPE_2E0,type,
c_2Ebool_2EDATATYPE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2EDATATYPE_2E1,type,
c_2Ebool_2EDATATYPE_2E1:
!>[A_27a: $tType] : ( A_27a > tyop_2Emin_2Ebool ) ).
tff(c_2Epred__set_2EDELETE_2E0,type,
c_2Epred__set_2EDELETE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EDELETE_2E2,type,
c_2Epred__set_2EDELETE_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * A_27a ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Epred__set_2EDIFF_2E0,type,
c_2Epred__set_2EDIFF_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EDIFF_2E2,type,
c_2Epred__set_2EDIFF_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Epred__set_2EDISJOINT_2E0,type,
c_2Epred__set_2EDISJOINT_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Epred__set_2EDISJOINT_2E2,type,
c_2Epred__set_2EDISJOINT_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Epred__set_2EEMPTY_2E0,type,
c_2Epred__set_2EEMPTY_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).
tff(c_2Earithmetic_2EEVEN_2E0,type,
c_2Earithmetic_2EEVEN_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).
tff(c_2Earithmetic_2EEVEN_2E1,type,
c_2Earithmetic_2EEVEN_2E1: tyop_2Enum_2Enum > tyop_2Emin_2Ebool ).
tff(c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E0,type,
c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal)) ).
tff(c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2,type,
c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Eextreal_2Eextreal ) ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
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_2Epair_2EFST_2E0,type,
c_2Epair_2EFST_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a) ).
tff(c_2Epair_2EFST_2E1,type,
c_2Epair_2EFST_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > A_27a ) ).
tff(c_2Epred__set_2EGSPEC_2E0,type,
c_2Epred__set_2EGSPEC_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Epred__set_2EGSPEC_2E1,type,
c_2Epred__set_2EGSPEC_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Ecombin_2EI_2E0,type,
c_2Ecombin_2EI_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(c_2Ecombin_2EI_2E1,type,
c_2Ecombin_2EI_2E1:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
tff(c_2Epred__set_2EIMAGE_2E0,type,
c_2Epred__set_2EIMAGE_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EIMAGE_2E2,type,
c_2Epred__set_2EIMAGE_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) ) ).
tff(c_2Ebool_2EIN_2E0,type,
c_2Ebool_2EIN_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2EIN_2E2,type,
c_2Ebool_2EIN_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Epred__set_2EINJ_2E0,type,
c_2Epred__set_2EINJ_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EINJ_2E3,type,
c_2Epred__set_2EINJ_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Epred__set_2EINSERT_2E0,type,
c_2Epred__set_2EINSERT_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EINSERT_2E2,type,
c_2Epred__set_2EINSERT_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Epred__set_2EINTER_2E0,type,
c_2Epred__set_2EINTER_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EINTER_2E2,type,
c_2Epred__set_2EINTER_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Epred__set_2EITSET_2E0,type,
c_2Epred__set_2EITSET_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,A_27b))) ).
tff(c_2Epred__set_2EITSET_2E3,type,
c_2Epred__set_2EITSET_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * A_27b ) > A_27b ) ).
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_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_2Eextreal_2ENegInf_2E0,type,
c_2Eextreal_2ENegInf_2E0: tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2ENormal_2E0,type,
c_2Eextreal_2ENormal_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal) ).
tff(c_2Eextreal_2ENormal_2E1,type,
c_2Eextreal_2ENormal_2E1: tyop_2Erealax_2Ereal > tyop_2Eextreal_2Eextreal ).
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_2Eextreal_2EPosInf_2E0,type,
c_2Eextreal_2EPosInf_2E0: tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2EQ__set_2E0,type,
c_2Eextreal_2EQ__set_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool) ).
tff(c_2Ereal__sigma_2EREAL__SUM__IMAGE_2E0,type,
c_2Ereal__sigma_2EREAL__SUM__IMAGE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal__sigma_2EREAL__SUM__IMAGE_2E2,type,
c_2Ereal__sigma_2EREAL__SUM__IMAGE_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Erealax_2Ereal ) ).
tff(c_2Epair_2ESND_2E0,type,
c_2Epair_2ESND_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b) ).
tff(c_2Epair_2ESND_2E1,type,
c_2Epair_2ESND_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > A_27b ) ).
tff(c_2Epred__set_2ESUBSET_2E0,type,
c_2Epred__set_2ESUBSET_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Epred__set_2ESUBSET_2E2,type,
c_2Epred__set_2ESUBSET_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > 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_2Ebool_2ETYPE__DEFINITION_2E0,type,
c_2Ebool_2ETYPE__DEFINITION_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
c_2Ebool_2ETYPE__DEFINITION_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Epair_2EUNCURRY_2E0,type,
c_2Epair_2EUNCURRY_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_2Epair_2Eprod(A_27a,A_27b),A_27c)) ).
tff(c_2Epair_2EUNCURRY_2E1,type,
c_2Epair_2EUNCURRY_2E1:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c) ) ).
tff(c_2Epred__set_2EUNION_2E0,type,
c_2Epred__set_2EUNION_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EUNION_2E2,type,
c_2Epred__set_2EUNION_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,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_2Erelation_2EWF_2E0,type,
c_2Erelation_2EWF_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool) ).
tff(c_2Erelation_2EWF_2E1,type,
c_2Erelation_2EWF_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Ebool ) ).
tff(c_2Erelation_2EWFREC_2E0,type,
c_2Erelation_2EWFREC_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,A_27b))) ).
tff(c_2Erelation_2EWFREC_2E2,type,
c_2Erelation_2EWFREC_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)) ) > tyop_2Emin_2Efun(A_27a,A_27b) ) ).
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_2Eextreal_2Eceiling_2E0,type,
c_2Eextreal_2Eceiling_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Enum_2Enum) ).
tff(c_2Eextreal_2Eceiling_2E1,type,
c_2Eextreal_2Eceiling_2E1: tyop_2Eextreal_2Eextreal > tyop_2Enum_2Enum ).
tff(c_2Epred__set_2Ecount_2E0,type,
c_2Epred__set_2Ecount_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Epred__set_2Ecount_2E1,type,
c_2Epred__set_2Ecount_2E1: tyop_2Enum_2Enum > tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).
tff(c_2Epred__set_2Ecountable_2E0,type,
c_2Epred__set_2Ecountable_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Epred__set_2Ecountable_2E1,type,
c_2Epred__set_2Ecountable_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Etransc_2Eexp_2E0,type,
c_2Etransc_2Eexp_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Etransc_2Eexp_2E1,type,
c_2Etransc_2Eexp_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Eextreal_2Eext__mono__decreasing_2E0,type,
c_2Eextreal_2Eext__mono__decreasing_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Ebool) ).
tff(c_2Eextreal_2Eext__mono__decreasing_2E1,type,
c_2Eextreal_2Eext__mono__decreasing_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal) > tyop_2Emin_2Ebool ).
tff(c_2Eextreal_2Eext__mono__increasing_2E0,type,
c_2Eextreal_2Eext__mono__increasing_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Ebool) ).
tff(c_2Eextreal_2Eext__mono__increasing_2E1,type,
c_2Eextreal_2Eext__mono__increasing_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal) > tyop_2Emin_2Ebool ).
tff(c_2Eextreal_2Eext__suminf_2E0,type,
c_2Eextreal_2Eext__suminf_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Eextreal_2Eextreal) ).
tff(c_2Eextreal_2Eext__suminf_2E1,type,
c_2Eextreal_2Eext__suminf_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal) > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__CASE_2E0,type,
c_2Eextreal_2Eextreal__CASE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a),A_27a)))) ).
tff(c_2Eextreal_2Eextreal__CASE_2E4,type,
c_2Eextreal_2Eextreal__CASE_2E4:
!>[A_27a: $tType] : ( ( tyop_2Eextreal_2Eextreal * A_27a * A_27a * tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a) ) > A_27a ) ).
tff(c_2Eextreal_2Eextreal__abs_2E0,type,
c_2Eextreal_2Eextreal__abs_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal) ).
tff(c_2Eextreal_2Eextreal__abs_2E1,type,
c_2Eextreal_2Eextreal__abs_2E1: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__add_2E0,type,
c_2Eextreal_2Eextreal__add_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)) ).
tff(c_2Eextreal_2Eextreal__add_2E2,type,
c_2Eextreal_2Eextreal__add_2E2: ( tyop_2Eextreal_2Eextreal * tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__ainv_2E0,type,
c_2Eextreal_2Eextreal__ainv_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal) ).
tff(c_2Eextreal_2Eextreal__ainv_2E1,type,
c_2Eextreal_2Eextreal__ainv_2E1: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__div_2E0,type,
c_2Eextreal_2Eextreal__div_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)) ).
tff(c_2Eextreal_2Eextreal__div_2E2,type,
c_2Eextreal_2Eextreal__div_2E2: ( tyop_2Eextreal_2Eextreal * tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__exp_2E0,type,
c_2Eextreal_2Eextreal__exp_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal) ).
tff(c_2Eextreal_2Eextreal__exp_2E1,type,
c_2Eextreal_2Eextreal__exp_2E1: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__inf_2E0,type,
c_2Eextreal_2Eextreal__inf_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal) ).
tff(c_2Eextreal_2Eextreal__inf_2E1,type,
c_2Eextreal_2Eextreal__inf_2E1: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool) > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__inv_2E0,type,
c_2Eextreal_2Eextreal__inv_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal) ).
tff(c_2Eextreal_2Eextreal__inv_2E1,type,
c_2Eextreal_2Eextreal__inv_2E1: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__le_2E0,type,
c_2Eextreal_2Eextreal__le_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)) ).
tff(c_2Eextreal_2Eextreal__le_2E2,type,
c_2Eextreal_2Eextreal__le_2E2: ( tyop_2Eextreal_2Eextreal * tyop_2Eextreal_2Eextreal ) > tyop_2Emin_2Ebool ).
tff(c_2Eextreal_2Eextreal__lg_2E0,type,
c_2Eextreal_2Eextreal__lg_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal) ).
tff(c_2Eextreal_2Eextreal__lg_2E1,type,
c_2Eextreal_2Eextreal__lg_2E1: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__logr_2E0,type,
c_2Eextreal_2Eextreal__logr_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)) ).
tff(c_2Eextreal_2Eextreal__logr_2E2,type,
c_2Eextreal_2Eextreal__logr_2E2: ( tyop_2Erealax_2Ereal * tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__lt_2E0,type,
c_2Eextreal_2Eextreal__lt_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)) ).
tff(c_2Eextreal_2Eextreal__lt_2E2,type,
c_2Eextreal_2Eextreal__lt_2E2: ( tyop_2Eextreal_2Eextreal * tyop_2Eextreal_2Eextreal ) > tyop_2Emin_2Ebool ).
tff(c_2Eextreal_2Eextreal__max_2E0,type,
c_2Eextreal_2Eextreal__max_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)) ).
tff(c_2Eextreal_2Eextreal__max_2E2,type,
c_2Eextreal_2Eextreal__max_2E2: ( tyop_2Eextreal_2Eextreal * tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__min_2E0,type,
c_2Eextreal_2Eextreal__min_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)) ).
tff(c_2Eextreal_2Eextreal__min_2E2,type,
c_2Eextreal_2Eextreal__min_2E2: ( tyop_2Eextreal_2Eextreal * tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__mul_2E0,type,
c_2Eextreal_2Eextreal__mul_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)) ).
tff(c_2Eextreal_2Eextreal__mul_2E2,type,
c_2Eextreal_2Eextreal__mul_2E2: ( tyop_2Eextreal_2Eextreal * tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__of__num_2E0,type,
c_2Eextreal_2Eextreal__of__num_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal) ).
tff(c_2Eextreal_2Eextreal__of__num_2E1,type,
c_2Eextreal_2Eextreal__of__num_2E1: tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__pow_2E0,type,
c_2Eextreal_2Eextreal__pow_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)) ).
tff(c_2Eextreal_2Eextreal__pow_2E2,type,
c_2Eextreal_2Eextreal__pow_2E2: ( tyop_2Eextreal_2Eextreal * tyop_2Enum_2Enum ) > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__size_2E0,type,
c_2Eextreal_2Eextreal__size_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Enum_2Enum) ).
tff(c_2Eextreal_2Eextreal__size_2E1,type,
c_2Eextreal_2Eextreal__size_2E1: tyop_2Eextreal_2Eextreal > tyop_2Enum_2Enum ).
tff(c_2Eextreal_2Eextreal__sqrt_2E0,type,
c_2Eextreal_2Eextreal__sqrt_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal) ).
tff(c_2Eextreal_2Eextreal__sqrt_2E1,type,
c_2Eextreal_2Eextreal__sqrt_2E1: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__sub_2E0,type,
c_2Eextreal_2Eextreal__sub_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)) ).
tff(c_2Eextreal_2Eextreal__sub_2E2,type,
c_2Eextreal_2Eextreal__sub_2E2: ( tyop_2Eextreal_2Eextreal * tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ).
tff(c_2Eextreal_2Eextreal__sup_2E0,type,
c_2Eextreal_2Eextreal__sup_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal) ).
tff(c_2Eextreal_2Eextreal__sup_2E1,type,
c_2Eextreal_2Eextreal__sup_2E1: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool) > tyop_2Eextreal_2Eextreal ).
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_2Eutil__prob_2Elogr_2E0,type,
c_2Eutil__prob_2Elogr_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Eutil__prob_2Elogr_2E2,type,
c_2Eutil__prob_2Elogr_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Eextreal_2Emono__decreasing_2E0,type,
c_2Eextreal_2Emono__decreasing_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool) ).
tff(c_2Eextreal_2Emono__decreasing_2E1,type,
c_2Eextreal_2Emono__decreasing_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Emin_2Ebool ).
tff(c_2Eextreal_2Emono__increasing_2E0,type,
c_2Eextreal_2Emono__increasing_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool) ).
tff(c_2Eextreal_2Emono__increasing_2E1,type,
c_2Eextreal_2Emono__increasing_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Emin_2Ebool ).
tff(c_2Ecombin_2Eo_2E0,type,
c_2Ecombin_2Eo_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,A_27b))) ).
tff(c_2Ecombin_2Eo_2E2,type,
c_2Ecombin_2Eo_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27c,A_27b) * tyop_2Emin_2Efun(A_27a,A_27c) ) > tyop_2Emin_2Efun(A_27a,A_27b) ) ).
tff(c_2Eextreal_2Eopen__interval_2E0,type,
c_2Eextreal_2Eopen__interval_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))) ).
tff(c_2Eextreal_2Eopen__interval_2E2,type,
c_2Eextreal_2Eopen__interval_2E2: ( tyop_2Eextreal_2Eextreal * tyop_2Eextreal_2Eextreal ) > tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool) ).
tff(c_2Eextreal_2Eopen__intervals__set_2E0,type,
c_2Eextreal_2Eopen__intervals__set_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
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_2Eextreal_2Erational__intervals_2E0,type,
c_2Eextreal_2Erational__intervals_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Eextreal_2Ereal_2E0,type,
c_2Eextreal_2Ereal_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Erealax_2Ereal) ).
tff(c_2Eextreal_2Ereal_2E1,type,
c_2Eextreal_2Ereal_2E1: tyop_2Eextreal_2Eextreal > 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_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_2Etransc_2Esqrt_2E0,type,
c_2Etransc_2Esqrt_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Etransc_2Esqrt_2E1,type,
c_2Etransc_2Esqrt_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Eseq_2Esuminf_2E0,type,
c_2Eseq_2Esuminf_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal) ).
tff(c_2Eseq_2Esuminf_2E1,type,
c_2Eseq_2Esuminf_2E1: 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_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_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: A_27b] : ( c_2Epair_2E_2C_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Epair_2E_2C_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Eextreal_2Eextreal_20mono_2Etyop_2Emin_2Ebool,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Eextreal_2Eextreal_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Emin_2Ebool,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eseq_2E_2D_2D_3E_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Eseq_2E_2D_2D_3E_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Eseq_2E_2D_2D_3E_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_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(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Eextreal_2Eextreal_2Ctyop_2Emin_2Efun_28tyop_2Eextreal_2Eextreal_2Ctyop_2Emin_2Ebool_29_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),c_2Emin_2E_40_2E0(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))),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(arityeq1_2Ec_2Epred__set_2ECARD_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECARD_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Epred__set_2ECARD_2E0(A_27a),X0_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Eextreal_2Eextreal,X2_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)),c_2Ebool_2ECOND_2E0(tyop_2Eextreal_2Eextreal),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(arityeq3_2Ec_2Eind__type_2ECONSTR_2E3_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal))] : ( c_2Eind__type_2ECONSTR_2E3(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal)),tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal)),tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal)),tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal))),c_2Eind__type_2ECONSTR_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ECROSS_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECROSS_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epred__set_2ECROSS_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ECROSS_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECROSS_2E2(A_27b,A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool)),c_2Epred__set_2ECROSS_2E0(A_27b,A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ECROSS_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECROSS_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),c_2Epred__set_2ECROSS_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2EDATATYPE_2E1_2Emono_2Etyop_2Emin_2Ebool,axiom,
! [X0_2E0: tyop_2Emin_2Ebool] : ( c_2Ebool_2EDATATYPE_2E1(tyop_2Emin_2Ebool,X0_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2EDATATYPE_2E0(tyop_2Emin_2Ebool),X0_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EDELETE_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: A_27a] : ( c_2Epred__set_2EDELETE_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EDELETE_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EDIFF_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDIFF_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EDIFF_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EDISJOINT_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDISJOINT_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EDISJOINT_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EEVEN_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEVEN_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,c_2Earithmetic_2EEVEN_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E0(A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),X1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)] : ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Eextreal_2Eextreal),X1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool)] : ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Epair_2Eprod(A_27b,A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E0(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Eextreal_2Eextreal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Eextreal_2Eextreal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EFST_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EFST_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a,c_2Epair_2EFST_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Eextreal_2Eextreal_20mono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Eextreal_2Eextreal_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Epair_2Eprod_28tyop_2Eextreal_2Eextreal_2Ctyop_2Eextreal_2Eextreal_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ecombin_2EI_2E1_2Emono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Ecombin_2EI_2E1(tyop_2Eextreal_2Eextreal,X0_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,c_2Ecombin_2EI_2E0(tyop_2Eextreal_2Eextreal),X0_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2EA_27a_20mono_2Etyop_2Eextreal_2Eextreal,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(A_27a,tyop_2Eextreal_2Eextreal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(A_27a,tyop_2Eextreal_2Eextreal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Eextreal_2Eextreal_20mono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,A_27a),X1_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Eextreal_2Eextreal,A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Eextreal_2Eextreal,A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Eextreal_2Eextreal_20mono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),X1_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Eextreal_2Eextreal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Epred__set_2EINJ_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINJ_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epred__set_2EINJ_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINSERT_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EINSERT_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EINTER_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINTER_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EINTER_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Epred__set_2EITSET_2E3_2Emono_2EA_27a_20mono_2Etyop_2Eextreal_2Eextreal,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X2_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Epred__set_2EITSET_2E3(A_27a,tyop_2Eextreal_2Eextreal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)),c_2Epred__set_2EITSET_2E0(A_27a,tyop_2Eextreal_2Eextreal),X0_2E0),X1_2E0),X2_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(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_2Eextreal_2ENormal_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2ENormal_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal,c_2Eextreal_2ENormal_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(arityeq2_2Ec_2Ereal__sigma_2EREAL__SUM__IMAGE_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ereal__sigma_2EREAL__SUM__IMAGE_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),c_2Ereal__sigma_2EREAL__SUM__IMAGE_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2ESND_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2ESND_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b,c_2Epair_2ESND_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2ESUBSET_2E0(A_27a),X0_2E0),X1_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_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Eind__type_2Erecspace_28tyop_2Erealax_2Ereal_29_20mono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),tyop_2Eextreal_2Eextreal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool),c_2Ebool_2ETYPE__DEFINITION_2E0(tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),tyop_2Eextreal_2Eextreal),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2Etyop_2Eextreal_2Eextreal,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal))] : ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Eextreal_2Eextreal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,tyop_2Eextreal_2Eextreal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27b_20mono_2EA_27a_20mono_2Etyop_2Eextreal_2Eextreal,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal))] : ( c_2Epair_2EUNCURRY_2E1(A_27b,A_27a,tyop_2Eextreal_2Eextreal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Eextreal_2Eextreal),c_2Epair_2EUNCURRY_2E0(A_27b,A_27a,tyop_2Eextreal_2Eextreal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2Etyop_2Eextreal_2Eextreal_20mono_2Etyop_2Eextreal_2Eextreal_20mono_2Etyop_2Epair_2Eprod_28tyop_2Emin_2Efun_28tyop_2Eextreal_2Eextreal_2Ctyop_2Emin_2Ebool_29_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(tyop_2Eextreal_2Eextreal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(tyop_2Eextreal_2Eextreal),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2EWF_2E1_2Emono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))] : ( c_2Erelation_2EWF_2E1(tyop_2Eextreal_2Eextreal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2EWF_2E0(tyop_2Eextreal_2Eextreal),X0_2E0) ) ).
tff(arityeq2_2Ec_2Erelation_2EWFREC_2E2_2Emono_2Etyop_2Eextreal_2Eextreal_20mono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal))] : ( c_2Erelation_2EWFREC_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)),c_2Erelation_2EWFREC_2E0(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
| p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).
tff(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_2Eextreal_2Eceiling_2E1,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eceiling_2E1(X0_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Enum_2Enum,c_2Eextreal_2Eceiling_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2Ecount_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Epred__set_2Ecount_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2Ecount_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2Ecountable_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2Ecountable_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2Ecountable_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2Ecountable_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2Ecountable_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2Ecountable_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2Ecountable_2E1_2Emono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2Ecountable_2E1(tyop_2Eextreal_2Eextreal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2Ecountable_2E0(tyop_2Eextreal_2Eextreal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2Ecountable_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Eextreal_2Eextreal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2Ecountable_2E1(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2Ecountable_2E0(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2Ecountable_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2Ecountable_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2Ecountable_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2Ecountable_2E1_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Enum_2Enum_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2Ecountable_2E1(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2Ecountable_2E0(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Etransc_2Eexp_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eexp_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Eexp_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eext__mono__decreasing_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] : ( c_2Eextreal_2Eext__mono__decreasing_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Ebool,c_2Eextreal_2Eext__mono__decreasing_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eext__mono__increasing_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] : ( c_2Eextreal_2Eext__mono__increasing_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Ebool,c_2Eextreal_2Eext__mono__increasing_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eext__suminf_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] : ( c_2Eextreal_2Eext__suminf_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eext__suminf_2E0,X0_2E0) ) ).
tff(arityeq4_2Ec_2Eextreal_2Eextreal__CASE_2E4_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: A_27a,X2_2E0: A_27a,X3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a)] : ( c_2Eextreal_2Eextreal__CASE_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a),A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a),A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a),A_27a)),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a),A_27a))),c_2Eextreal_2Eextreal__CASE_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq4_2Ec_2Eextreal_2Eextreal__CASE_2E4_2Emono_2Etyop_2Eextreal_2Eextreal,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Eextreal_2Eextreal,X2_2E0: tyop_2Eextreal_2Eextreal,X3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal)] : ( c_2Eextreal_2Eextreal__CASE_2E4(tyop_2Eextreal_2Eextreal,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal),tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal),tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal),tyop_2Eextreal_2Eextreal)),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal),tyop_2Eextreal_2Eextreal))),c_2Eextreal_2Eextreal__CASE_2E0(tyop_2Eextreal_2Eextreal),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eextreal__abs_2E1,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__abs_2E1(X0_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__abs_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2Eextreal__add_2E2,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),c_2Eextreal_2Eextreal__add_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eextreal__ainv_2E1,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__ainv_2E1(X0_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__ainv_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2Eextreal__div_2E2,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__div_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),c_2Eextreal_2Eextreal__div_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eextreal__exp_2E1,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__exp_2E1(X0_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__exp_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eextreal__inf_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] : ( c_2Eextreal_2Eextreal__inf_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__inf_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eextreal__inv_2E1,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__inv_2E1(X0_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__inv_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2Eextreal__le_2E2,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__le_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),c_2Eextreal_2Eextreal__le_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eextreal__lg_2E1,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__lg_2E1(X0_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__lg_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2Eextreal__logr_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__logr_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),c_2Eextreal_2Eextreal__logr_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2Eextreal__lt_2E2,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__lt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),c_2Eextreal_2Eextreal__lt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2Eextreal__max_2E2,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__max_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),c_2Eextreal_2Eextreal__max_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2Eextreal__min_2E2,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__min_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),c_2Eextreal_2Eextreal__min_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2Eextreal__mul_2E2,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),c_2Eextreal_2Eextreal__mul_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eextreal__of__num_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__of__num_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__of__num_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2Eextreal__pow_2E2,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__pow_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),c_2Eextreal_2Eextreal__pow_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eextreal__size_2E1,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__size_2E1(X0_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Enum_2Enum,c_2Eextreal_2Eextreal__size_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eextreal__sqrt_2E1,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__sqrt_2E1(X0_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__sqrt_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2Eextreal__sub_2E2,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__sub_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),c_2Eextreal_2Eextreal__sub_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Eextreal__sup_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] : ( c_2Eextreal_2Eextreal__sup_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__sup_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_2Eutil__prob_2Elogr_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Eutil__prob_2Elogr_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_2Eutil__prob_2Elogr_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Emono__decreasing_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eextreal_2Emono__decreasing_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Eextreal_2Emono__decreasing_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eextreal_2Emono__increasing_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eextreal_2Emono__increasing_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Eextreal_2Emono__increasing_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Eextreal_2Eextreal_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Eextreal_2Eextreal,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Eextreal_2Eextreal,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eextreal_2Eopen__interval_2E2,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal,X1_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eopen__interval_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),c_2Eextreal_2Eopen__interval_2E0,X0_2E0),X1_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(arityeq1_2Ec_2Eextreal_2Ereal_2E1,axiom,
! [X0_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Ereal_2E1(X0_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Erealax_2Ereal,c_2Eextreal_2Ereal_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__add_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__add_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__lt_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Erealax_2Ereal__lt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_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(arityeq1_2Ec_2Etransc_2Esqrt_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Esqrt_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Esqrt_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eseq_2Esuminf_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Esuminf_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Eseq_2Esuminf_2E0,X0_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_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_2Eextreal_2Emono__decreasing__def,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eextreal_2Emono__decreasing_2E1(V0f_2E0))
<=> ! [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(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1m_2E0))) ) ) ).
tff(thm_2Eextreal_2Emono__increasing__def,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eextreal_2Emono__increasing_2E1(V0f_2E0))
<=> ! [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(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0))) ) ) ).
tff(thm_2Eextreal_2Eextreal__sqrt__def,axiom,
( ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2Eextreal__sqrt_2E1(c_2Eextreal_2ENormal_2E1(V0x_2E0)) = c_2Eextreal_2ENormal_2E1(c_2Etransc_2Esqrt_2E1(V0x_2E0)) )
& ( c_2Eextreal_2Eextreal__sqrt_2E1(c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2EPosInf_2E0 ) ) ).
tff(thm_2Eextreal_2Eextreal__pow__def,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V3n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F1_2E0,V3n_2E0))
<=> ( V3n_2E0 = c_2Enum_2E0_2E0 ) )
=> ( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2n_2E0))
<=> ( V2n_2E0 = c_2Enum_2E0_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__pow_2E2(c_2Eextreal_2ENormal_2E1(V0a_2E0),V1n_2E0) = c_2Eextreal_2ENormal_2E1(c_2Ereal_2Epow_2E2(V0a_2E0,V1n_2E0)) )
& ! [V2n_2E0: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__pow_2E2(c_2Eextreal_2EPosInf_2E0,V2n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2n_2E0),c_2Eextreal_2ENormal_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Eextreal_2EPosInf_2E0) )
& ! [V3n_2E0: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__pow_2E2(c_2Eextreal_2ENegInf_2E0,V3n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F1_2E0,V3n_2E0),c_2Eextreal_2ENormal_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,c_2Earithmetic_2EEVEN_2E1(V3n_2E0),c_2Eextreal_2EPosInf_2E0,c_2Eextreal_2ENegInf_2E0)) ) ) ) ) ).
tff(thm_2Eextreal_2Eextreal__exp__def,axiom,
( ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2Eextreal__exp_2E1(c_2Eextreal_2ENormal_2E1(V0x_2E0)) = c_2Eextreal_2ENormal_2E1(c_2Etransc_2Eexp_2E1(V0x_2E0)) )
& ( c_2Eextreal_2Eextreal__exp_2E1(c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__exp_2E1(c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2ENormal_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ).
tff(thm_2Eextreal_2Eextreal__lg__def,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__lg_2E1(V0x_2E0) = c_2Eextreal_2Eextreal__logr_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0) ) ).
tff(thm_2Eextreal_2Eextreal__logr__def,axiom,
( ! [V0b_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2Eextreal__logr_2E2(V0b_2E0,c_2Eextreal_2ENormal_2E1(V1x_2E0)) = c_2Eextreal_2ENormal_2E1(c_2Eutil__prob_2Elogr_2E2(V0b_2E0,V1x_2E0)) )
& ! [V2b_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2Eextreal__logr_2E2(V2b_2E0,c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ! [V3b_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2Eextreal__logr_2E2(V3b_2E0,c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2ENegInf_2E0 ) ) ).
tff(thm_2Eextreal_2Eextreal__abs__primitive__def,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal))] :
( ! [V0R_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))] : ( app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,F0_2E0,V0R_2E0) = c_2Erelation_2EWF_2E1(tyop_2Eextreal_2Eextreal,V0R_2E0) )
=> ( ! [V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal,F2_2E0,V3x_2E0) = c_2Ecombin_2EI_2E1(tyop_2Eextreal_2Eextreal,c_2Eextreal_2ENormal_2E1(c_2Ereal_2Eabs_2E1(V3x_2E0))) )
=> ( ! [V1extreal__abs_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),V2a_2E0: tyop_2Eextreal_2Eextreal] : ( app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),F1_2E0,V1extreal__abs_2E0),V2a_2E0) = c_2Eextreal_2Eextreal__CASE_2E4(tyop_2Eextreal_2Eextreal,V2a_2E0,c_2Ecombin_2EI_2E1(tyop_2Eextreal_2Eextreal,c_2Eextreal_2EPosInf_2E0),c_2Ecombin_2EI_2E1(tyop_2Eextreal_2Eextreal,c_2Eextreal_2EPosInf_2E0),F2_2E0) )
=> ( c_2Eextreal_2Eextreal__abs_2E0 = c_2Erelation_2EWFREC_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,c_2Emin_2E_40_2E1(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),F0_2E0),F1_2E0) ) ) ) ) ).
tff(thm_2Eextreal_2Eextreal__div__def,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__div_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,c_2Eextreal_2Eextreal__inv_2E1(V1y_2E0)) ) ).
tff(thm_2Eextreal_2Eextreal__ainv__def,axiom,
( ( c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2ENegInf_2E0 )
& ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2ENormal_2E1(V0x_2E0)) = c_2Eextreal_2ENormal_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) ) ) ).
tff(thm_2Eextreal_2Eextreal__inv__def,axiom,
( ( c_2Eextreal_2Eextreal__inv_2E1(c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2ENormal_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
& ( c_2Eextreal_2Eextreal__inv_2E1(c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2ENormal_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
& ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2Eextreal__inv_2E1(c_2Eextreal_2ENormal_2E1(V0x_2E0)) = c_2Eextreal_2ENormal_2E1(c_2Erealax_2Einv_2E1(V0x_2E0)) ) ) ).
tff(thm_2Eextreal_2Eextreal__lt__def,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0))
<=> ~ p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Eextreal_2Ereal__def,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F1_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,F0_2E0,V0x_2E0))
<=> ( ( V0x_2E0 = c_2Eextreal_2ENegInf_2E0 )
| ( V0x_2E0 = c_2Eextreal_2EPosInf_2E0 ) ) )
=> ( ! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1r_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0x_2E0),V1r_2E0))
<=> ( V0x_2E0 = c_2Eextreal_2ENormal_2E1(V1r_2E0) ) )
=> ! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Ereal_2E1(V0x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,F0_2E0,V0x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0x_2E0))) ) ) ) ).
tff(thm_2Eextreal_2Eextreal__of__num__def,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__of__num_2E1(V0n_2E0) = c_2Eextreal_2ENormal_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)) ) ).
tff(thm_2Eextreal_2Eextreal__TY__DEF,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal)),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal)),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal)),F0_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)] :
( ! [V7n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),F3_2E0,V7n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Erealax_2Ereal) )
=> ( ! [V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),F2_2E0,V5n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Erealax_2Ereal) )
=> ( ! [V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),F1_2E0,V4n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Erealax_2Ereal) )
=> ( ! [V1a0_2E0: tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal)] :
( p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,F0_2E0,V1a0_2E0))
<=> ! [V2_27extreal_27_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)] :
( ! [V3a0_2E0: tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal)] :
( ( ( V3a0_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Erealax_2Ereal,c_2Enum_2E0_2E0,c_2Ebool_2EARB_2E0(tyop_2Erealax_2Ereal),F1_2E0) )
| ( V3a0_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Erealax_2Ereal,c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0),c_2Ebool_2EARB_2E0(tyop_2Erealax_2Ereal),F2_2E0) )
| ? [V6a_2E0: tyop_2Erealax_2Ereal] : ( V3a0_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Erealax_2Ereal,c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0)),V6a_2E0,F3_2E0) ) )
=> p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V2_27extreal_27_2E0,V3a0_2E0)) )
=> p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V2_27extreal_27_2E0,V1a0_2E0)) ) )
=> ? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Eind__type_2Erecspace(tyop_2Erealax_2Ereal),tyop_2Eextreal_2Eextreal,F0_2E0,V0rep_2E0)) ) ) ) ) ).
tff(thm_2Eextreal_2Eextreal__case__def,axiom,
! [A_27a: $tType] :
( ! [V0v_2E0: A_27a,V1v1_2E0: A_27a,V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a)] : ( c_2Eextreal_2Eextreal__CASE_2E4(A_27a,c_2Eextreal_2ENegInf_2E0,V0v_2E0,V1v1_2E0,V2f_2E0) = V0v_2E0 )
& ! [V3v_2E0: A_27a,V4v1_2E0: A_27a,V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a)] : ( c_2Eextreal_2Eextreal__CASE_2E4(A_27a,c_2Eextreal_2EPosInf_2E0,V3v_2E0,V4v1_2E0,V5f_2E0) = V4v1_2E0 )
& ! [V6a_2E0: tyop_2Erealax_2Ereal,V7v_2E0: A_27a,V8v1_2E0: A_27a,V9f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a)] : ( c_2Eextreal_2Eextreal__CASE_2E4(A_27a,c_2Eextreal_2ENormal_2E1(V6a_2E0),V7v_2E0,V8v1_2E0,V9f_2E0) = app_2E2(tyop_2Erealax_2Ereal,A_27a,V9f_2E0,V6a_2E0) ) ) ).
tff(thm_2Eextreal_2Eextreal__size__def,axiom,
( ( c_2Eextreal_2Eextreal__size_2E1(c_2Eextreal_2ENegInf_2E0) = c_2Enum_2E0_2E0 )
& ( c_2Eextreal_2Eextreal__size_2E1(c_2Eextreal_2EPosInf_2E0) = c_2Enum_2E0_2E0 )
& ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2Eextreal__size_2E1(c_2Eextreal_2ENormal_2E1(V0a_2E0)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ) ) ).
tff(thm_2Eextreal_2Eext__mono__increasing__def,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( p(c_2Eextreal_2Eext__mono__increasing_2E1(V0f_2E0))
<=> ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V1m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0))) ) ) ).
tff(thm_2Eextreal_2Eext__mono__decreasing__def,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( p(c_2Eextreal_2Eext__mono__decreasing_2E1(V0f_2E0))
<=> ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V1m_2E0))) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__DEF,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2e_2E0: A_27a,V3acc_2E0: tyop_2Eextreal_2Eextreal] : ( app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)),F0_2E0,V0f_2E0),V2e_2E0),V3acc_2E0) = c_2Eextreal_2Eextreal__add_2E2(app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V2e_2E0),V3acc_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,V1s_2E0) = c_2Epred__set_2EITSET_2E3(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal)),F0_2E0,V0f_2E0),V1s_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ).
tff(thm_2Eextreal_2Eextreal__sup__def,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V0p_2E0))
<=> ! [V1x_2E0: tyop_2Eextreal_2Eextreal] :
( ! [V2y_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V2y_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V2y_2E0,V1x_2E0)) )
=> ( V1x_2E0 = c_2Eextreal_2EPosInf_2E0 ) ) )
=> ( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F1_2E0,V0p_2E0))
<=> ! [V3x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V3x_2E0))
=> ( V3x_2E0 = c_2Eextreal_2ENegInf_2E0 ) ) )
=> ( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V4r_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F2_2E0,V0p_2E0),V4r_2E0) = app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,c_2Eextreal_2ENormal_2E1(V4r_2E0)) )
=> ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] : ( c_2Eextreal_2Eextreal__sup_2E1(V0p_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V0p_2E0),c_2Eextreal_2EPosInf_2E0,c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F1_2E0,V0p_2E0),c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2ENormal_2E1(c_2Ereal_2Esup_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F2_2E0,V0p_2E0))))) ) ) ) ) ).
tff(thm_2Eextreal_2Eextreal__inf__def,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] : ( c_2Eextreal_2Eextreal__inf_2E1(V0p_2E0) = c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__ainv_2E0,V0p_2E0))) ) ).
tff(thm_2Eextreal_2Eext__suminf__def,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F0_2E0,V0f_2E0),V1n_2E0) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Enum_2Enum,V0f_2E0,c_2Epred__set_2Ecount_2E1(V1n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] : ( c_2Eextreal_2Eext__suminf_2E1(V0f_2E0) = c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F0_2E0,V0f_2E0),c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) ) ) ).
tff(thm_2Eextreal_2Eextreal__min__def,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__min_2E2(V0x_2E0,V1y_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0),V0x_2E0,V1y_2E0) ) ).
tff(thm_2Eextreal_2Eextreal__max__def,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__max_2E2(V0x_2E0,V1y_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0),V1y_2E0,V0x_2E0) ) ).
tff(thm_2Eextreal_2EQ__set__def,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),F2_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F1_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,F3_2E0,V0x_2E0))
<=> ? [V1a_2E0: tyop_2Enum_2Enum,V2b_2E0: tyop_2Enum_2Enum] :
( ( V0x_2E0 = c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(V1a_2E0),c_2Eextreal_2Eextreal__of__num_2E1(V2b_2E0)) )
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__of__num_2E1(V2b_2E0))) ) )
=> ( ! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0x_2E0,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,F3_2E0,V0x_2E0)) )
=> ( ! [V3x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,F2_2E0,V3x_2E0))
<=> ? [V4a_2E0: tyop_2Enum_2Enum,V5b_2E0: tyop_2Enum_2Enum] :
( ( V3x_2E0 = c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(V4a_2E0),c_2Eextreal_2Eextreal__of__num_2E1(V5b_2E0))) )
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__of__num_2E1(V5b_2E0))) ) )
=> ( ! [V3x_2E0: tyop_2Eextreal_2Eextreal] : ( app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F1_2E0,V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,F2_2E0,V3x_2E0)) )
=> ( c_2Eextreal_2EQ__set_2E0 = c_2Epred__set_2EUNION_2E2(tyop_2Eextreal_2Eextreal,c_2Epred__set_2EGSPEC_2E1(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,F0_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,F1_2E0)) ) ) ) ) ) ).
tff(thm_2Eextreal_2Eceiling__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_2Eextreal_2Eceiling_2E1(c_2Eextreal_2ENormal_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_2Eextreal_2Eopen__interval__def,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Eextreal_2Eextreal,V2x_2E0: tyop_2Eextreal_2Eextreal,V1b_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V2x_2E0),V1b_2E0))
<=> ( p(c_2Eextreal_2Eextreal__lt_2E2(V0a_2E0,V2x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V2x_2E0,V1b_2E0)) ) )
=> ( ! [V0a_2E0: tyop_2Eextreal_2Eextreal,V1b_2E0: tyop_2Eextreal_2Eextreal,V2x_2E0: tyop_2Eextreal_2Eextreal] : ( app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V2x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Eextreal_2Eextreal,V1b_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eopen__interval_2E2(V0a_2E0,V1b_2E0) = c_2Epred__set_2EGSPEC_2E1(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0)) ) ) ) ).
tff(thm_2Eextreal_2Eopen__intervals__set__def,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Eextreal_2Eextreal,V1b_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F1_2E0,V0a_2E0),V1b_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V0a_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Eextreal_2Eextreal)))
& p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V1b_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Eextreal_2Eextreal))) ) )
=> ( ! [V0a_2E0: tyop_2Eextreal_2Eextreal,V1b_2E0: tyop_2Eextreal_2Eextreal] : ( app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1b_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Eextreal_2Eopen__interval_2E2(V0a_2E0,V1b_2E0),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F1_2E0,V0a_2E0),V1b_2E0)) )
=> ( c_2Eextreal_2Eopen__intervals__set_2E0 = c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),c_2Epair_2EUNCURRY_2E1(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0)) ) ) ) ).
tff(thm_2Eextreal_2Erational__intervals__def,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Eextreal_2Eextreal,V1b_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F1_2E0,V0a_2E0),V1b_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V0a_2E0,c_2Eextreal_2EQ__set_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V1b_2E0,c_2Eextreal_2EQ__set_2E0)) ) )
=> ( ! [V0a_2E0: tyop_2Eextreal_2Eextreal,V1b_2E0: tyop_2Eextreal_2Eextreal] : ( app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1b_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Eextreal_2Eopen__interval_2E2(V0a_2E0,V1b_2E0),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F1_2E0,V0a_2E0),V1b_2E0)) )
=> ( c_2Eextreal_2Erational__intervals_2E0 = c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal),c_2Epair_2EUNCURRY_2E1(tyop_2Eextreal_2Eextreal,tyop_2Eextreal_2Eextreal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0)) ) ) ) ).
tff(thm_2Eextreal_2Enum__not__infty,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( ( c_2Eextreal_2Eextreal__of__num_2E1(V0n_2E0) != c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__of__num_2E1(V0n_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) ).
tff(thm_2Eextreal_2Eextreal__not__infty,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Eextreal_2ENormal_2E1(V0x_2E0) != c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2ENormal_2E1(V0x_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) ).
tff(thm_2Eextreal_2Eextreal__eq__zero,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Eextreal_2ENormal_2E1(V0x_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Eextreal_2Eextreal__cases,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( V0x_2E0 = c_2Eextreal_2ENegInf_2E0 )
| ( V0x_2E0 = c_2Eextreal_2EPosInf_2E0 )
| ? [V1r_2E0: tyop_2Erealax_2Ereal] : ( V0x_2E0 = c_2Eextreal_2ENormal_2E1(V1r_2E0) ) ) ).
tff(thm_2Eextreal_2Emono__increasing__converges__to__sup,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1r_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eextreal_2Emono__increasing_2E1(V0f_2E0))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V1r_2E0)) )
=> ( V1r_2E0 = c_2Ereal_2Esup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) ) ) ).
tff(thm_2Eextreal_2Emono__decreasing__suc,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eextreal_2Emono__decreasing_2E1(V0f_2E0))
<=> ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0))) ) ).
tff(thm_2Eextreal_2Emono__increasing__suc,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eextreal_2Emono__increasing_2E1(V0f_2E0))
<=> ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)))) ) ).
tff(thm_2Eextreal_2ELOGR__MONO__LE__IMP,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2b_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(V0x_2E0,V1y_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))),V2b_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Eutil__prob_2Elogr_2E2(V2b_2E0,V0x_2E0),c_2Eutil__prob_2Elogr_2E2(V2b_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2ELOGR__MONO__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2b_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_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V2b_2E0)) )
=> ( c_2Ereal_2Ereal__lte_2E2(c_2Eutil__prob_2Elogr_2E2(V2b_2E0,V0x_2E0),c_2Eutil__prob_2Elogr_2E2(V2b_2E0,V1y_2E0)) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0) ) ) ).
tff(thm_2Eextreal_2EPOW__NEG__ODD,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V1x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Epow_2E2(V1x_2E0,V0n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Earithmetic_2EODD_2E1(V0n_2E0) ) ) ).
tff(thm_2Eextreal_2EPOW__POS__EVEN,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V1x_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),c_2Ereal_2Epow_2E2(V1x_2E0,V0n_2E0)) = c_2Earithmetic_2EEVEN_2E1(V0n_2E0) ) ) ).
tff(thm_2Eextreal_2EREAL__LE__RDIV__EQ__NEG,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_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
=> ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2E_2F_2E2(V1y_2E0,V2z_2E0),V0x_2E0) = c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0),V1y_2E0) ) ) ).
tff(thm_2Eextreal_2EREAL__LT__RDIV__EQ__NEG,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_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(V1y_2E0,V2z_2E0),V0x_2E0) = c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0),V1y_2E0) ) ) ).
tff(thm_2Eextreal_2EREAL__LT__RMUL__NEG__0__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(V1y_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)) ) ).
tff(thm_2Eextreal_2EREAL__LT__LMUL__NEG__0__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(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),V1y_2E0)) ) ).
tff(thm_2Eextreal_2EREAL__LT__RMUL__NEG__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_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),V1y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Eextreal_2EREAL__LT__LMUL__NEG__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_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(V1y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Eextreal_2EREAL__LT__RMUL__0__NEG,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),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) )
=> p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Eextreal_2EREAL__LT__LMUL__0__NEG,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),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) )
=> p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Eextreal_2Eextreal__abs__def,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Eextreal_2Eextreal__abs_2E1(c_2Eextreal_2ENormal_2E1(V0x_2E0)) = c_2Eextreal_2ENormal_2E1(c_2Ereal_2Eabs_2E1(V0x_2E0)) )
& ( c_2Eextreal_2Eextreal__abs_2E1(c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__abs_2E1(c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2EPosInf_2E0 ) ) ).
tff(thm_2Eextreal_2Eextreal__abs__ind,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] :
( ( ! [V1x_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0P_2E0,c_2Eextreal_2ENormal_2E1(V1x_2E0)))
& p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0P_2E0,c_2Eextreal_2ENegInf_2E0))
& p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0P_2E0,c_2Eextreal_2EPosInf_2E0)) )
=> ! [V2v_2E0: tyop_2Eextreal_2Eextreal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0P_2E0,V2v_2E0)) ) ).
tff(thm_2Eextreal_2Eextreal__mul__def,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),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,F3_2E0,V0y_2E0))
<=> ( V0y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F2_2E0,V1x_2E0))
<=> ( V1x_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,F1_2E0,V0y_2E0))
<=> ( V0y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V1x_2E0))
<=> ( V1x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2EPosInf_2E0,c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2EPosInf_2E0,c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V1x_2E0),c_2Eextreal_2ENegInf_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V1x_2E0),c_2Eextreal_2ENormal_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0),c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2EPosInf_2E0)) )
& ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2ENormal_2E1(V0y_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0y_2E0),c_2Eextreal_2ENormal_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0y_2E0),c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2EPosInf_2E0)) )
& ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V1x_2E0),c_2Eextreal_2EPosInf_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F2_2E0,V1x_2E0),c_2Eextreal_2ENormal_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0),c_2Eextreal_2EPosInf_2E0,c_2Eextreal_2ENegInf_2E0)) )
& ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2EPosInf_2E0,c_2Eextreal_2ENormal_2E1(V0y_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F3_2E0,V0y_2E0),c_2Eextreal_2ENormal_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0y_2E0),c_2Eextreal_2EPosInf_2E0,c_2Eextreal_2ENegInf_2E0)) )
& ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V1x_2E0),c_2Eextreal_2ENormal_2E1(V0y_2E0)) = c_2Eextreal_2ENormal_2E1(c_2Erealax_2Ereal__mul_2E2(V1x_2E0,V0y_2E0)) ) ) ) ) ) ) ).
tff(thm_2Eextreal_2Eextreal__mul__ind,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))] :
( ( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENegInf_2E0),c_2Eextreal_2ENegInf_2E0))
& p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENegInf_2E0),c_2Eextreal_2EPosInf_2E0))
& p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2EPosInf_2E0),c_2Eextreal_2ENegInf_2E0))
& p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2EPosInf_2E0),c_2Eextreal_2EPosInf_2E0))
& ! [V1x_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENormal_2E1(V1x_2E0)),c_2Eextreal_2ENegInf_2E0))
& ! [V2y_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENegInf_2E0),c_2Eextreal_2ENormal_2E1(V2y_2E0)))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENormal_2E1(V3x_2E0)),c_2Eextreal_2EPosInf_2E0))
& ! [V4y_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2EPosInf_2E0),c_2Eextreal_2ENormal_2E1(V4y_2E0)))
& ! [V5x_2E0: tyop_2Erealax_2Ereal,V6y_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENormal_2E1(V5x_2E0)),c_2Eextreal_2ENormal_2E1(V6y_2E0))) )
=> ! [V7v_2E0: tyop_2Eextreal_2Eextreal,V8v1_2E0: tyop_2Eextreal_2Eextreal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,V7v_2E0),V8v1_2E0)) ) ).
tff(thm_2Eextreal_2Eextreal__le__def,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2v5_2E0: tyop_2Erealax_2Ereal,V3v3_2E0: tyop_2Erealax_2Ereal,V4v2_2E0: tyop_2Erealax_2Ereal,V5a_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2ENormal_2E1(V1x_2E0),c_2Eextreal_2ENormal_2E1(V0y_2E0)) = c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V0y_2E0) )
& ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2ENegInf_2E0,V5a_2E0) = c_2Ebool_2ET_2E0 )
& ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2EPosInf_2E0,c_2Eextreal_2EPosInf_2E0) = c_2Ebool_2ET_2E0 )
& ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2ENormal_2E1(V4v2_2E0),c_2Eextreal_2EPosInf_2E0) = c_2Ebool_2ET_2E0 )
& ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2EPosInf_2E0,c_2Eextreal_2ENegInf_2E0) = c_2Ebool_2EF_2E0 )
& ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2ENormal_2E1(V3v3_2E0),c_2Eextreal_2ENegInf_2E0) = c_2Ebool_2EF_2E0 )
& ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2EPosInf_2E0,c_2Eextreal_2ENormal_2E1(V2v5_2E0)) = c_2Ebool_2EF_2E0 ) ) ).
tff(thm_2Eextreal_2Eextreal__le__ind,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))] :
( ( ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENormal_2E1(V1x_2E0)),c_2Eextreal_2ENormal_2E1(V2y_2E0)))
& ! [V3a_2E0: tyop_2Eextreal_2Eextreal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENegInf_2E0),V3a_2E0))
& p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2EPosInf_2E0),c_2Eextreal_2EPosInf_2E0))
& ! [V4v2_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENormal_2E1(V4v2_2E0)),c_2Eextreal_2EPosInf_2E0))
& p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2EPosInf_2E0),c_2Eextreal_2ENegInf_2E0))
& ! [V5v3_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENormal_2E1(V5v3_2E0)),c_2Eextreal_2ENegInf_2E0))
& ! [V6v5_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2EPosInf_2E0),c_2Eextreal_2ENormal_2E1(V6v5_2E0))) )
=> ! [V7v_2E0: tyop_2Eextreal_2Eextreal,V8v1_2E0: tyop_2Eextreal_2Eextreal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,V7v_2E0),V8v1_2E0)) ) ).
tff(thm_2Eextreal_2Eextreal__sub__def,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2v5_2E0: tyop_2Erealax_2Ereal,V3v3_2E0: tyop_2Erealax_2Ereal,V4v2_2E0: tyop_2Erealax_2Ereal,V5a_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2ENormal_2E1(V1x_2E0),c_2Eextreal_2ENormal_2E1(V0y_2E0)) = c_2Eextreal_2ENormal_2E1(c_2Ereal_2Ereal__sub_2E2(V1x_2E0,V0y_2E0)) )
& ( c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2EPosInf_2E0,V5a_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2ENormal_2E1(V4v2_2E0),c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2ENormal_2E1(V2v5_2E0)) = c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2ENormal_2E1(V3v3_2E0),c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2EPosInf_2E0 ) ) ).
tff(thm_2Eextreal_2Eextreal__sub__ind,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))] :
( ( ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENormal_2E1(V1x_2E0)),c_2Eextreal_2ENormal_2E1(V2y_2E0)))
& ! [V3a_2E0: tyop_2Eextreal_2Eextreal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2EPosInf_2E0),V3a_2E0))
& p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENegInf_2E0),c_2Eextreal_2EPosInf_2E0))
& ! [V4v2_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENormal_2E1(V4v2_2E0)),c_2Eextreal_2EPosInf_2E0))
& p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENegInf_2E0),c_2Eextreal_2ENegInf_2E0))
& ! [V5v5_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENegInf_2E0),c_2Eextreal_2ENormal_2E1(V5v5_2E0)))
& ! [V6v3_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENormal_2E1(V6v3_2E0)),c_2Eextreal_2ENegInf_2E0)) )
=> ! [V7v_2E0: tyop_2Eextreal_2Eextreal,V8v1_2E0: tyop_2Eextreal_2Eextreal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,V7v_2E0),V8v1_2E0)) ) ).
tff(thm_2Eextreal_2Eextreal__add__def,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2v5_2E0: tyop_2Erealax_2Ereal,V3v3_2E0: tyop_2Erealax_2Ereal,V4v2_2E0: tyop_2Erealax_2Ereal,V5a_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2ENormal_2E1(V1x_2E0),c_2Eextreal_2ENormal_2E1(V0y_2E0)) = c_2Eextreal_2ENormal_2E1(c_2Erealax_2Ereal__add_2E2(V1x_2E0,V0y_2E0)) )
& ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2EPosInf_2E0,V5a_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2ENormal_2E1(V4v2_2E0),c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2ENormal_2E1(V2v5_2E0)) = c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2ENormal_2E1(V3v3_2E0),c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2ENegInf_2E0 ) ) ).
tff(thm_2Eextreal_2Eextreal__add__ind,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))] :
( ( ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENormal_2E1(V1x_2E0)),c_2Eextreal_2ENormal_2E1(V2y_2E0)))
& ! [V3a_2E0: tyop_2Eextreal_2Eextreal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2EPosInf_2E0),V3a_2E0))
& p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENegInf_2E0),c_2Eextreal_2EPosInf_2E0))
& ! [V4v2_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENormal_2E1(V4v2_2E0)),c_2Eextreal_2EPosInf_2E0))
& p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENegInf_2E0),c_2Eextreal_2ENegInf_2E0))
& ! [V5v5_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENegInf_2E0),c_2Eextreal_2ENormal_2E1(V5v5_2E0)))
& ! [V6v3_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,c_2Eextreal_2ENormal_2E1(V6v3_2E0)),c_2Eextreal_2ENegInf_2E0)) )
=> ! [V7v_2E0: tyop_2Eextreal_2Eextreal,V8v1_2E0: tyop_2Eextreal_2Eextreal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V0P_2E0,V7v_2E0),V8v1_2E0)) ) ).
tff(thm_2Eextreal_2Enormal__real,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2ENormal_2E1(c_2Eextreal_2Ereal_2E1(V0x_2E0)) = V0x_2E0 ) ) ).
tff(thm_2Eextreal_2Ereal__normal,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2Ereal_2E1(c_2Eextreal_2ENormal_2E1(V0x_2E0)) = V0x_2E0 ) ).
tff(thm_2Eextreal_2Eextreal__case__eq,axiom,
! [A_27a: $tType,V0x_2E0: tyop_2Eextreal_2Eextreal,V1v1_2E0: A_27a,V2v_27_2E0: A_27a,V3v_2E0: A_27a,V4f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a)] :
( ( c_2Eextreal_2Eextreal__CASE_2E4(A_27a,V0x_2E0,V3v_2E0,V1v1_2E0,V4f_2E0) = V2v_27_2E0 )
<=> ( ( ( V0x_2E0 = c_2Eextreal_2ENegInf_2E0 )
& ( V3v_2E0 = V2v_27_2E0 ) )
| ( ( V0x_2E0 = c_2Eextreal_2EPosInf_2E0 )
& ( V1v1_2E0 = V2v_27_2E0 ) )
| ? [V5r_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 = c_2Eextreal_2ENormal_2E1(V5r_2E0) )
& ( app_2E2(tyop_2Erealax_2Ereal,A_27a,V4f_2E0,V5r_2E0) = V2v_27_2E0 ) ) ) ) ).
tff(thm_2Eextreal_2Eextreal__case__cong,axiom,
! [A_27a: $tType,V0v1_27_2E0: A_27a,V1v_27_2E0: A_27a,V2f_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a),V3M_2E0: tyop_2Eextreal_2Eextreal,V4M_27_2E0: tyop_2Eextreal_2Eextreal,V5v_2E0: A_27a,V6v1_2E0: A_27a,V7f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a)] :
( ( ( V3M_2E0 = V4M_27_2E0 )
& ( ( V4M_27_2E0 = c_2Eextreal_2ENegInf_2E0 )
=> ( V5v_2E0 = V1v_27_2E0 ) )
& ( ( V4M_27_2E0 = c_2Eextreal_2EPosInf_2E0 )
=> ( V6v1_2E0 = V0v1_27_2E0 ) )
& ! [V8a_2E0: tyop_2Erealax_2Ereal] :
( ( V4M_27_2E0 = c_2Eextreal_2ENormal_2E1(V8a_2E0) )
=> ( app_2E2(tyop_2Erealax_2Ereal,A_27a,V7f_2E0,V8a_2E0) = app_2E2(tyop_2Erealax_2Ereal,A_27a,V2f_27_2E0,V8a_2E0) ) ) )
=> ( c_2Eextreal_2Eextreal__CASE_2E4(A_27a,V3M_2E0,V5v_2E0,V6v1_2E0,V7f_2E0) = c_2Eextreal_2Eextreal__CASE_2E4(A_27a,V4M_27_2E0,V1v_27_2E0,V0v1_27_2E0,V2f_27_2E0) ) ) ).
tff(thm_2Eextreal_2Eextreal__induction,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] :
( ( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0P_2E0,c_2Eextreal_2ENegInf_2E0))
& p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0P_2E0,c_2Eextreal_2EPosInf_2E0))
& ! [V1r_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0P_2E0,c_2Eextreal_2ENormal_2E1(V1r_2E0))) )
=> ! [V2e_2E0: tyop_2Eextreal_2Eextreal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0P_2E0,V2e_2E0)) ) ).
tff(thm_2Eextreal_2Eextreal__Axiom,axiom,
! [A_27a: $tType,V0f0_2E0: A_27a,V1f1_2E0: A_27a,V2f2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,A_27a)] :
? [V3fn_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,A_27a)] :
( ( app_2E2(tyop_2Eextreal_2Eextreal,A_27a,V3fn_2E0,c_2Eextreal_2ENegInf_2E0) = V0f0_2E0 )
& ( app_2E2(tyop_2Eextreal_2Eextreal,A_27a,V3fn_2E0,c_2Eextreal_2EPosInf_2E0) = V1f1_2E0 )
& ! [V4a_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Eextreal_2Eextreal,A_27a,V3fn_2E0,c_2Eextreal_2ENormal_2E1(V4a_2E0)) = app_2E2(tyop_2Erealax_2Ereal,A_27a,V2f2_2E0,V4a_2E0) ) ) ).
tff(thm_2Eextreal_2Eextreal__nchotomy,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( V0x_2E0 = c_2Eextreal_2ENegInf_2E0 )
| ( V0x_2E0 = c_2Eextreal_2EPosInf_2E0 )
| ? [V1r_2E0: tyop_2Erealax_2Ereal] : ( V0x_2E0 = c_2Eextreal_2ENormal_2E1(V1r_2E0) ) ) ).
tff(thm_2Eextreal_2Eextreal__distinct,axiom,
( ( c_2Eextreal_2ENegInf_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2ENegInf_2E0 != c_2Eextreal_2ENormal_2E1(V0a_2E0) )
& ! [V1a_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2EPosInf_2E0 != c_2Eextreal_2ENormal_2E1(V1a_2E0) ) ) ).
tff(thm_2Eextreal_2Edatatype__extreal,axiom,
! [V0extreal_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Ebool)))] : p(c_2Ebool_2EDATATYPE_2E1(tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Ebool),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Ebool)),V0extreal_2E0,c_2Eextreal_2ENegInf_2E0),c_2Eextreal_2EPosInf_2E0),c_2Eextreal_2ENormal_2E0))) ).
tff(thm_2Eextreal_2Eextreal__11,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1a_27_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Eextreal_2ENormal_2E1(V0a_2E0) = c_2Eextreal_2ENormal_2E1(V1a_27_2E0) )
<=> ( V0a_2E0 = V1a_27_2E0 ) ) ).
tff(thm_2Eextreal_2Emul__rzero,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Eextreal_2Emul__lzero,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Eextreal_2Emul__rone,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = V0x_2E0 ) ).
tff(thm_2Eextreal_2Emul__lone,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextreal_2Eentire,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V0x_2E0 = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V1y_2E0 = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Eextreal_2Eextreal__lt__eq,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2ENormal_2E1(V0x_2E0),c_2Eextreal_2ENormal_2E1(V1y_2E0)) = c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Eextreal_2Ele__refl,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V0x_2E0)) ).
tff(thm_2Eextreal_2Elt__refl,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ~ p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V0x_2E0)) ).
tff(thm_2Eextreal_2Ele__infty,axiom,
( ! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2ENegInf_2E0,V0x_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,c_2Eextreal_2EPosInf_2E0)) )
& ! [V1x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(V1x_2E0,c_2Eextreal_2ENegInf_2E0))
<=> ( V1x_2E0 = c_2Eextreal_2ENegInf_2E0 ) )
& ! [V2x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2EPosInf_2E0,V2x_2E0))
<=> ( V2x_2E0 = c_2Eextreal_2EPosInf_2E0 ) ) ) ).
tff(thm_2Eextreal_2Elt__infty,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2ENormal_2E1(V1y_2E0)))
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2ENormal_2E1(V1y_2E0),c_2Eextreal_2EPosInf_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2EPosInf_2E0))
& ~ p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,c_2Eextreal_2ENegInf_2E0))
& ~ p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2EPosInf_2E0,V0x_2E0))
& ( ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
<=> p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,c_2Eextreal_2EPosInf_2E0)) )
& ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
<=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2ENegInf_2E0,V0x_2E0)) ) ) ).
tff(thm_2Eextreal_2Elt__imp__le,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Eextreal_2Elt__imp__ne,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0))
=> ( V0x_2E0 != V1y_2E0 ) ) ).
tff(thm_2Eextreal_2Ele__trans,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Eextreal_2Elt__trans,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Eextreal_2Elet__trans,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Eextreal_2Ele__antisym,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,V0x_2E0)) )
<=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Eextreal_2Elt__antisym,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
~ ( p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Eextreal_2Elte__trans,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V2z_2E0)) ) ).
tff(thm_2Eextreal_2Ele__total,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0))
| p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Eextreal_2Elt__total,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( V0x_2E0 = V1y_2E0 )
| p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0))
| p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Eextreal_2Ele__01,axiom,
p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ).
tff(thm_2Eextreal_2Elt__01,axiom,
p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ).
tff(thm_2Eextreal_2Ene__01,axiom,
c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) != c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Eextreal_2Ele__02,axiom,
p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ).
tff(thm_2Eextreal_2Elt__02,axiom,
p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ).
tff(thm_2Eextreal_2Ene__02,axiom,
c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) != c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Eextreal_2Ele__num,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__of__num_2E1(V0n_2E0))) ).
tff(thm_2Eextreal_2Elt__le,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0))
<=> ( p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0))
& ( V0x_2E0 != V1y_2E0 ) ) ) ).
tff(thm_2Eextreal_2Ele__lt,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0))
<=> ( p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0))
| ( V0x_2E0 = V1y_2E0 ) ) ) ).
tff(thm_2Eextreal_2Ele__neg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__ainv_2E1(V0x_2E0),c_2Eextreal_2Eextreal__ainv_2E1(V1y_2E0)) = c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Eextreal_2Elt__neg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__ainv_2E1(V0x_2E0),c_2Eextreal_2Eextreal__ainv_2E1(V1y_2E0)) = c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Eextreal_2Ele__add,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Elt__add,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Elet__add,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Elte__add,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Ele__add2,axiom,
! [V0w_2E0: tyop_2Eextreal_2Eextreal,V1x_2E0: tyop_2Eextreal_2Eextreal,V2y_2E0: tyop_2Eextreal_2Eextreal,V3z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(V0w_2E0,V1x_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V2y_2E0,V3z_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__add_2E2(V0w_2E0,V2y_2E0),c_2Eextreal_2Eextreal__add_2E2(V1x_2E0,V3z_2E0))) ) ).
tff(thm_2Eextreal_2Elt__add2,axiom,
! [V0w_2E0: tyop_2Eextreal_2Eextreal,V1x_2E0: tyop_2Eextreal_2Eextreal,V2y_2E0: tyop_2Eextreal_2Eextreal,V3z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(V0w_2E0,V1x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V2y_2E0,V3z_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__add_2E2(V0w_2E0,V2y_2E0),c_2Eextreal_2Eextreal__add_2E2(V1x_2E0,V3z_2E0))) ) ).
tff(thm_2Eextreal_2Elet__add2,axiom,
! [V0w_2E0: tyop_2Eextreal_2Eextreal,V1x_2E0: tyop_2Eextreal_2Eextreal,V2y_2E0: tyop_2Eextreal_2Eextreal,V3z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0w_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0w_2E0 != c_2Eextreal_2EPosInf_2E0 )
& p(c_2Eextreal_2Eextreal__le_2E2(V0w_2E0,V1x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V2y_2E0,V3z_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__add_2E2(V0w_2E0,V2y_2E0),c_2Eextreal_2Eextreal__add_2E2(V1x_2E0,V3z_2E0))) ) ).
tff(thm_2Eextreal_2Elet__add2__alt,axiom,
! [V0w_2E0: tyop_2Eextreal_2Eextreal,V1x_2E0: tyop_2Eextreal_2Eextreal,V2y_2E0: tyop_2Eextreal_2Eextreal,V3z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V1x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1x_2E0 != c_2Eextreal_2EPosInf_2E0 )
& p(c_2Eextreal_2Eextreal__le_2E2(V0w_2E0,V1x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V2y_2E0,V3z_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__add_2E2(V0w_2E0,V2y_2E0),c_2Eextreal_2Eextreal__add_2E2(V1x_2E0,V3z_2E0))) ) ).
tff(thm_2Eextreal_2Ele__addr,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0) ) ) ).
tff(thm_2Eextreal_2Ele__addr__imp,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Ele__ladd,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,V2z_2E0) ) ) ).
tff(thm_2Eextreal_2Ele__radd,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V0x_2E0),c_2Eextreal_2Eextreal__add_2E2(V2z_2E0,V0x_2E0)) = c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,V2z_2E0) ) ) ).
tff(thm_2Eextreal_2Ele__radd__imp,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,V2z_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V0x_2E0),c_2Eextreal_2Eextreal__add_2E2(V2z_2E0,V0x_2E0))) ) ).
tff(thm_2Eextreal_2Ele__ladd__imp,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,V2z_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V2z_2E0))) ) ).
tff(thm_2Eextreal_2Elt__ladd,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,V2z_2E0) ) ) ).
tff(thm_2Eextreal_2Elt__radd,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V0x_2E0),c_2Eextreal_2Eextreal__add_2E2(V2z_2E0,V0x_2E0)) = c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,V2z_2E0) ) ) ).
tff(thm_2Eextreal_2Elt__addl,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) ) ) ).
tff(thm_2Eextreal_2Ele__lneg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__ainv_2E1(V0x_2E0),V1y_2E0) = c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Eextreal_2Ele__mul,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Elet__mul,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Elte__mul,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Ele__mul__neg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)))
& p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Emul__le,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Eextreal_2Elt__mul,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Elt__mul__neg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)))
& p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Emul__lt,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Eextreal_2Emul__let,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Eextreal_2Emul__lte,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Eextreal_2Ele__rmul__imp,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V2z_2E0),c_2Eextreal_2Eextreal__mul_2E2(V1y_2E0,V2z_2E0))) ) ).
tff(thm_2Eextreal_2Ele__lmul__imp,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__mul_2E2(V2z_2E0,V0x_2E0),c_2Eextreal_2Eextreal__mul_2E2(V2z_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Elt__rmul,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
& ( V2z_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V2z_2E0),c_2Eextreal_2Eextreal__mul_2E2(V1y_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0) ) ) ).
tff(thm_2Eextreal_2Elt__lmul,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,V2z_2E0) ) ) ).
tff(thm_2Eextreal_2Elt__mul2,axiom,
! [V0x1_2E0: tyop_2Eextreal_2Eextreal,V1x2_2E0: tyop_2Eextreal_2Eextreal,V2y1_2E0: tyop_2Eextreal_2Eextreal,V3y2_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x1_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V2y1_2E0))
& ( V0x1_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V2y1_2E0 != c_2Eextreal_2EPosInf_2E0 )
& p(c_2Eextreal_2Eextreal__lt_2E2(V0x1_2E0,V1x2_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V2y1_2E0,V3y2_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x1_2E0,V2y1_2E0),c_2Eextreal_2Eextreal__mul_2E2(V1x2_2E0,V3y2_2E0))) ) ).
tff(thm_2Eextreal_2Eabs__pos,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__abs_2E1(V0x_2E0))) ).
tff(thm_2Eextreal_2Eabs__refl,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__abs_2E1(V0x_2E0) = V0x_2E0 )
<=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0)) ) ).
tff(thm_2Eextreal_2Eabs__bounds,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1k_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__abs_2E1(V0x_2E0),V1k_2E0))
<=> ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__ainv_2E1(V1k_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1k_2E0)) ) ) ).
tff(thm_2Eextreal_2Eabs__bounds__lt,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1k_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__abs_2E1(V0x_2E0),V1k_2E0))
<=> ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__ainv_2E1(V1k_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1k_2E0)) ) ) ).
tff(thm_2Eextreal_2Eadd__comm,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Eextreal_2Eadd__assoc,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0),V2z_2E0) ) ).
tff(thm_2Eextreal_2Eadd__not__infty,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0) != c_2Eextreal_2ENegInf_2E0 ) )
& ( ( ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) ) ).
tff(thm_2Eextreal_2Eadd__rzero,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)) = V0x_2E0 ) ).
tff(thm_2Eextreal_2Eadd__lzero,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextreal_2Eadd__infty,axiom,
( ! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2EPosInf_2E0,V0x_2E0) = c_2Eextreal_2EPosInf_2E0 ) )
& ! [V1x_2E0: tyop_2Eextreal_2Eextreal] :
( ( V1x_2E0 != c_2Eextreal_2EPosInf_2E0 )
=> ( ( c_2Eextreal_2Eextreal__add_2E2(V1x_2E0,c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2ENegInf_2E0,V1x_2E0) = c_2Eextreal_2ENegInf_2E0 ) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__EQ__LADD,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( ( c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V2z_2E0) )
<=> ( V1y_2E0 = V2z_2E0 ) ) ) ).
tff(thm_2Eextreal_2Esub__rzero,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)) = V0x_2E0 ) ).
tff(thm_2Eextreal_2Esub__lzero,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0) = c_2Eextreal_2Eextreal__ainv_2E1(V0x_2E0) ) ).
tff(thm_2Eextreal_2Esub__le__imp,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
& p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__add_2E2(V2z_2E0,V0x_2E0))) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0),V2z_2E0)) ) ).
tff(thm_2Eextreal_2Esub__le__imp2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 )
& p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__add_2E2(V2z_2E0,V0x_2E0))) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0),V2z_2E0)) ) ).
tff(thm_2Eextreal_2Ele__sub__imp,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V0x_2E0),V2z_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__sub_2E2(V2z_2E0,V0x_2E0))) ) ).
tff(thm_2Eextreal_2Elt__sub__imp,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V0x_2E0),V2z_2E0))
=> p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,c_2Eextreal_2Eextreal__sub_2E2(V2z_2E0,V0x_2E0))) ) ).
tff(thm_2Eextreal_2Esub__lt__imp,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
& p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,c_2Eextreal_2Eextreal__add_2E2(V2z_2E0,V0x_2E0))) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0),V2z_2E0)) ) ).
tff(thm_2Eextreal_2Esub__lt__imp2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V2z_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2EPosInf_2E0 )
& p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,c_2Eextreal_2Eextreal__add_2E2(V2z_2E0,V0x_2E0))) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0),V2z_2E0)) ) ).
tff(thm_2Eextreal_2Esub__zero__lt,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0))
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0))) ) ).
tff(thm_2Eextreal_2Esub__zero__lt2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0))) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Eextreal_2Esub__lt__zero,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0))
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Eextreal_2Esub__lt__zero2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 )
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Eextreal_2Esub__zero__le,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Eextreal_2Esub__le__zero,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ).
tff(thm_2Eextreal_2Ele__sub__eq,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__sub_2E2(V2z_2E0,V0x_2E0)) = c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V0x_2E0),V2z_2E0) ) ) ).
tff(thm_2Eextreal_2Ele__sub__eq2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V2z_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__sub_2E2(V2z_2E0,V0x_2E0)) = c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V0x_2E0),V2z_2E0) ) ) ).
tff(thm_2Eextreal_2Esub__le__eq,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0),V2z_2E0) = c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__add_2E2(V2z_2E0,V0x_2E0)) ) ) ).
tff(thm_2Eextreal_2Esub__le__eq2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2ENegInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0),V2z_2E0) = c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__add_2E2(V2z_2E0,V0x_2E0)) ) ) ).
tff(thm_2Eextreal_2Esub__le__switch,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0),V2z_2E0) = c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V2z_2E0),V0x_2E0) ) ) ).
tff(thm_2Eextreal_2Esub__le__switch2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0),V2z_2E0) = c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V2z_2E0),V0x_2E0) ) ) ).
tff(thm_2Eextreal_2Elt__sub,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V2z_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V0x_2E0),V2z_2E0) = c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,c_2Eextreal_2Eextreal__sub_2E2(V2z_2E0,V0x_2E0)) ) ) ).
tff(thm_2Eextreal_2Esub__add2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0)) = V1y_2E0 ) ) ).
tff(thm_2Eextreal_2Eadd__sub,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0),V1y_2E0) = V0x_2E0 ) ) ).
tff(thm_2Eextreal_2Eadd__sub2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V0x_2E0),V1y_2E0) = V0x_2E0 ) ) ).
tff(thm_2Eextreal_2Esub__add,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0),V1y_2E0) = V0x_2E0 ) ) ).
tff(thm_2Eextreal_2Eextreal__sub__add,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,c_2Eextreal_2Eextreal__ainv_2E1(V1y_2E0)) ) ).
tff(thm_2Eextreal_2Esub__0,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Eextreal_2Eneg__neg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2Eextreal__ainv_2E1(V0x_2E0)) = V0x_2E0 ) ).
tff(thm_2Eextreal_2Eneg__0,axiom,
c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Eextreal_2Eneg__eq0,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__ainv_2E1(V0x_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V0x_2E0 = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Eextreal_2Eeq__neg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__ainv_2E1(V0x_2E0) = c_2Eextreal_2Eextreal__ainv_2E1(V1y_2E0) )
<=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Eextreal_2Eneg__minus1,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__ainv_2E1(V0x_2E0) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0) ) ).
tff(thm_2Eextreal_2Esub__rneg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,c_2Eextreal_2Eextreal__ainv_2E1(V1y_2E0)) = c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Eextreal_2Esub__lneg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 ) )
| ( ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 ) ) )
=> ( c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2Eextreal__ainv_2E1(V0x_2E0),V1y_2E0) = c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0)) ) ) ).
tff(thm_2Eextreal_2Eneg__sub,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
| ( ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 ) ) )
=> ( c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0)) = c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V0x_2E0) ) ) ).
tff(thm_2Eextreal_2Esub__not__infty,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0) != c_2Eextreal_2ENegInf_2E0 ) )
& ( ( ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) ) ).
tff(thm_2Eextreal_2Ele__lsub__imp,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,V2z_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V2z_2E0),c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Eextreal_2Eeq__sub__ladd__normal,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 = c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,c_2Eextreal_2ENormal_2E1(V2z_2E0)) )
<=> ( c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,c_2Eextreal_2ENormal_2E1(V2z_2E0)) = V1y_2E0 ) ) ).
tff(thm_2Eextreal_2Eeq__sub__radd,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( ( c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0) = V2z_2E0 )
<=> ( V0x_2E0 = c_2Eextreal_2Eextreal__add_2E2(V2z_2E0,V1y_2E0) ) ) ) ).
tff(thm_2Eextreal_2Eeq__sub__ladd,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V2z_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( ( V0x_2E0 = c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V2z_2E0) )
<=> ( c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V2z_2E0) = V1y_2E0 ) ) ) ).
tff(thm_2Eextreal_2Eeq__sub__switch,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 = c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2ENormal_2E1(V2z_2E0),V1y_2E0) )
<=> ( V1y_2E0 = c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2ENormal_2E1(V2z_2E0),V0x_2E0) ) ) ).
tff(thm_2Eextreal_2Eeq__add__sub__switch,axiom,
! [V0a_2E0: tyop_2Eextreal_2Eextreal,V1b_2E0: tyop_2Eextreal_2Eextreal,V2c_2E0: tyop_2Eextreal_2Eextreal,V3d_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V1b_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1b_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V2c_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V2c_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( ( c_2Eextreal_2Eextreal__add_2E2(V0a_2E0,V1b_2E0) = c_2Eextreal_2Eextreal__add_2E2(V2c_2E0,V3d_2E0) )
<=> ( c_2Eextreal_2Eextreal__sub_2E2(V0a_2E0,V2c_2E0) = c_2Eextreal_2Eextreal__sub_2E2(V3d_2E0,V1b_2E0) ) ) ) ).
tff(thm_2Eextreal_2Esub__refl,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V0x_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Eextreal_2Emul__comm,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__mul_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Eextreal_2Emul__assoc,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,c_2Eextreal_2Eextreal__mul_2E2(V1y_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0),V2z_2E0) ) ).
tff(thm_2Eextreal_2Emul__not__infty,axiom,
( ! [V0c_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0c_2E0))
& ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V0c_2E0),V1y_2E0) != c_2Eextreal_2ENegInf_2E0 ) )
& ! [V2c_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2c_2E0))
& ( V3y_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V2c_2E0),V3y_2E0) != c_2Eextreal_2EPosInf_2E0 ) )
& ! [V4c_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V4c_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
& ( V5y_2E0 != c_2Eextreal_2ENegInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V4c_2E0),V5y_2E0) != c_2Eextreal_2EPosInf_2E0 ) )
& ! [V6c_2E0: tyop_2Erealax_2Ereal,V7y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V6c_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
& ( V7y_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V6c_2E0),V7y_2E0) != c_2Eextreal_2ENegInf_2E0 ) ) ) ).
tff(thm_2Eextreal_2Emul__not__infty2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0) != c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) ) ).
tff(thm_2Eextreal_2Eadd__ldistrib__pos,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0)) )
=> ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V2z_2E0)) ) ) ).
tff(thm_2Eextreal_2Eadd__ldistrib__neg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)))
& p(c_2Eextreal_2Eextreal__le_2E2(V2z_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) )
=> ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V2z_2E0)) ) ) ).
tff(thm_2Eextreal_2Eadd__ldistrib__normal,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
| ( ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2ENegInf_2E0 ) ) )
=> ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V0x_2E0),c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V0x_2E0),V1y_2E0),c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V0x_2E0),V2z_2E0)) ) ) ).
tff(thm_2Eextreal_2Eadd__ldistrib__normal2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V0x_2E0),c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V0x_2E0),V1y_2E0),c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V0x_2E0),V2z_2E0)) ) ) ).
tff(thm_2Eextreal_2Eadd__rdistrib__normal,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
| ( ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2ENegInf_2E0 ) ) )
=> ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V2z_2E0),c_2Eextreal_2ENormal_2E1(V0x_2E0)) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__mul_2E2(V1y_2E0,c_2Eextreal_2ENormal_2E1(V0x_2E0)),c_2Eextreal_2Eextreal__mul_2E2(V2z_2E0,c_2Eextreal_2ENormal_2E1(V0x_2E0))) ) ) ).
tff(thm_2Eextreal_2Eadd__rdistrib__normal2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V2z_2E0),c_2Eextreal_2ENormal_2E1(V0x_2E0)) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__mul_2E2(V1y_2E0,c_2Eextreal_2ENormal_2E1(V0x_2E0)),c_2Eextreal_2Eextreal__mul_2E2(V2z_2E0,c_2Eextreal_2ENormal_2E1(V0x_2E0))) ) ) ).
tff(thm_2Eextreal_2Eadd__ldistrib,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0)) )
| ( p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)))
& p(c_2Eextreal_2Eextreal__le_2E2(V2z_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) ) )
=> ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V2z_2E0)) ) ) ).
tff(thm_2Eextreal_2Eadd__rdistrib,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0)) )
| ( p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)))
& p(c_2Eextreal_2Eextreal__le_2E2(V2z_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0))) ) )
=> ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V2z_2E0),V0x_2E0) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__mul_2E2(V1y_2E0,V0x_2E0),c_2Eextreal_2Eextreal__mul_2E2(V2z_2E0,V0x_2E0)) ) ) ).
tff(thm_2Eextreal_2Emul__lneg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__ainv_2E1(V0x_2E0),V1y_2E0) = c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Eextreal_2Emul__rneg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,c_2Eextreal_2Eextreal__ainv_2E1(V1y_2E0)) = c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Eextreal_2Eneg__mul2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__ainv_2E1(V0x_2E0),c_2Eextreal_2Eextreal__ainv_2E1(V1y_2E0)) = c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Eextreal_2Eadd2__sub2,axiom,
! [V0a_2E0: tyop_2Eextreal_2Eextreal,V1b_2E0: tyop_2Eextreal_2Eextreal,V2c_2E0: tyop_2Eextreal_2Eextreal,V3d_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( ( V1b_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V3d_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
| ( ( V1b_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V3d_2E0 != c_2Eextreal_2ENegInf_2E0 ) ) )
=> ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__sub_2E2(V0a_2E0,V1b_2E0),c_2Eextreal_2Eextreal__sub_2E2(V2c_2E0,V3d_2E0)) = c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2Eextreal__add_2E2(V0a_2E0,V2c_2E0),c_2Eextreal_2Eextreal__add_2E2(V1b_2E0,V3d_2E0)) ) ) ).
tff(thm_2Eextreal_2Esub__ldistrib,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,c_2Eextreal_2Eextreal__sub_2E2(V1y_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V2z_2E0)) ) ) ).
tff(thm_2Eextreal_2Esub__rdistrib,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0),V2z_2E0) = c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V2z_2E0),c_2Eextreal_2Eextreal__mul_2E2(V1y_2E0,V2z_2E0)) ) ) ).
tff(thm_2Eextreal_2Eextreal__div__eq,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2ENormal_2E1(V0x_2E0),c_2Eextreal_2ENormal_2E1(V1y_2E0)) = c_2Eextreal_2ENormal_2E1(c_2Ereal_2E_2F_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Eextreal_2Einv__one,axiom,
c_2Eextreal_2Eextreal__inv_2E1(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Eextreal_2Einv__1over,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__inv_2E1(V0x_2E0) = c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0) ) ).
tff(thm_2Eextreal_2Ediv__one,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__div_2E2(V0x_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = V0x_2E0 ) ).
tff(thm_2Eextreal_2Einv__pos,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0))) ) ).
tff(thm_2Eextreal_2Erinv__uniq,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
=> ( V1y_2E0 = c_2Eextreal_2Eextreal__inv_2E1(V0x_2E0) ) ) ).
tff(thm_2Eextreal_2Elinv__uniq,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
=> ( V0x_2E0 = c_2Eextreal_2Eextreal__inv_2E1(V1y_2E0) ) ) ).
tff(thm_2Eextreal_2Ele__rdiv,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__mul_2E2(V1y_2E0,c_2Eextreal_2ENormal_2E1(V0x_2E0)),V2z_2E0) = c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__div_2E2(V2z_2E0,c_2Eextreal_2ENormal_2E1(V0x_2E0))) ) ) ).
tff(thm_2Eextreal_2Ele__ldiv,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__mul_2E2(V2z_2E0,c_2Eextreal_2ENormal_2E1(V0x_2E0))) = c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__div_2E2(V1y_2E0,c_2Eextreal_2ENormal_2E1(V0x_2E0)),V2z_2E0) ) ) ).
tff(thm_2Eextreal_2Elt__rdiv,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
=> ( c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,c_2Eextreal_2Eextreal__div_2E2(V1y_2E0,c_2Eextreal_2ENormal_2E1(V2z_2E0))) = c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,c_2Eextreal_2ENormal_2E1(V2z_2E0)),V1y_2E0) ) ) ).
tff(thm_2Eextreal_2Elt__ldiv,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
=> ( c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__div_2E2(V0x_2E0,c_2Eextreal_2ENormal_2E1(V2z_2E0)),V1y_2E0) = c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,c_2Eextreal_2Eextreal__mul_2E2(V1y_2E0,c_2Eextreal_2ENormal_2E1(V2z_2E0))) ) ) ).
tff(thm_2Eextreal_2Elt__rdiv__neg,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
=> ( c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__div_2E2(V1y_2E0,c_2Eextreal_2ENormal_2E1(V2z_2E0)),V0x_2E0) = c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,c_2Eextreal_2ENormal_2E1(V2z_2E0)),V1y_2E0) ) ) ).
tff(thm_2Eextreal_2Ediv__add,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal,V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1y_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V2z_2E0 != c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__div_2E2(V0x_2E0,V2z_2E0),c_2Eextreal_2Eextreal__div_2E2(V1y_2E0,V2z_2E0)) = c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0),V2z_2E0) ) ) ).
tff(thm_2Eextreal_2Ele__inv,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__inv_2E1(V0x_2E0))) ) ).
tff(thm_2Eextreal_2Epow__0,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,c_2Enum_2E0_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
tff(thm_2Eextreal_2Epow__1,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) = V0x_2E0 ) ).
tff(thm_2Eextreal_2Epow__2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V0x_2E0) ) ).
tff(thm_2Eextreal_2Epow__zero,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__pow_2E2(V1x_2E0,c_2Enum_2ESUC_2E1(V0n_2E0)) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V1x_2E0 = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Eextreal_2Epow__zero__imp,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__pow_2E2(V1x_2E0,V0n_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( V1x_2E0 = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Eextreal_2Ele__pow2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ).
tff(thm_2Eextreal_2Epow__pos__le,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,V1n_2E0))) ) ).
tff(thm_2Eextreal_2Epow__pos__lt,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,V1n_2E0))) ) ).
tff(thm_2Eextreal_2Epow__le,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Eextreal_2Eextreal,V2y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V1x_2E0,V2y_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__pow_2E2(V1x_2E0,V0n_2E0),c_2Eextreal_2Eextreal__pow_2E2(V2y_2E0,V0n_2E0))) ) ).
tff(thm_2Eextreal_2Epow__lt,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Eextreal_2Eextreal,V2y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V1x_2E0,V2y_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__pow_2E2(V1x_2E0,c_2Enum_2ESUC_2E1(V0n_2E0)),c_2Eextreal_2Eextreal__pow_2E2(V2y_2E0,c_2Enum_2ESUC_2E1(V0n_2E0)))) ) ).
tff(thm_2Eextreal_2Epow__lt2,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Eextreal_2Eextreal,V2y_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V0n_2E0 != c_2Enum_2E0_2E0 )
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V1x_2E0,V2y_2E0)) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__pow_2E2(V1x_2E0,V0n_2E0),c_2Eextreal_2Eextreal__pow_2E2(V2y_2E0,V0n_2E0))) ) ).
tff(thm_2Eextreal_2Epow__le__mono,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1n_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2m_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,V1n_2E0),c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,V2m_2E0))) ) ).
tff(thm_2Eextreal_2Epow__pos__even,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(V1x_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)))
=> ( c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__pow_2E2(V1x_2E0,V0n_2E0)) = c_2Earithmetic_2EEVEN_2E1(V0n_2E0) ) ) ).
tff(thm_2Eextreal_2Epow__neg__odd,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(V1x_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)))
=> ( c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__pow_2E2(V1x_2E0,V0n_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Earithmetic_2EODD_2E1(V0n_2E0) ) ) ).
tff(thm_2Eextreal_2Eadd__pow2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__pow_2E2(c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__pow_2E2(V1y_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0),V1y_2E0)) ) ).
tff(thm_2Eextreal_2Epow__add,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1n_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2m_2E0)) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,V1n_2E0),c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,V2m_2E0)) ) ).
tff(thm_2Eextreal_2Epow__mul,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Eextreal_2Eextreal,V2y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__pow_2E2(c_2Eextreal_2Eextreal__mul_2E2(V1x_2E0,V2y_2E0),V0n_2E0) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__pow_2E2(V1x_2E0,V0n_2E0),c_2Eextreal_2Eextreal__pow_2E2(V2y_2E0,V0n_2E0)) ) ).
tff(thm_2Eextreal_2Epow__minus1,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__pow_2E2(c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2Eextreal__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_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
tff(thm_2Eextreal_2Epow__not__infty,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Eextreal_2Eextreal] :
( ( ( V1x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V1x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( ( c_2Eextreal_2Eextreal__pow_2E2(V1x_2E0,V0n_2E0) != c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__pow_2E2(V1x_2E0,V0n_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) ) ).
tff(thm_2Eextreal_2Esqrt__pos__le,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__sqrt_2E1(V0x_2E0))) ) ).
tff(thm_2Eextreal_2Esqrt__pos__lt,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__sqrt_2E1(V0x_2E0))) ) ).
tff(thm_2Eextreal_2Epow2__sqrt,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Eextreal_2Eextreal__sqrt_2E1(c_2Eextreal_2Eextreal__pow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) = V0x_2E0 ) ) ).
tff(thm_2Eextreal_2Esqrt__pow2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__pow_2E2(c_2Eextreal_2Eextreal__sqrt_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = V0x_2E0 )
<=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0)) ) ).
tff(thm_2Eextreal_2Esqrt__mono__le,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sqrt_2E1(V0x_2E0),c_2Eextreal_2Eextreal__sqrt_2E1(V1y_2E0))) ) ).
tff(thm_2Eextreal_2Elogr__not__infty,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0x_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> ( ( c_2Eextreal_2Eextreal__logr_2E2(V1b_2E0,V0x_2E0) != c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__logr_2E2(V1b_2E0,V0x_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) ) ).
tff(thm_2Eextreal_2Ehalf__between,axiom,
( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))))
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Eextreal_2Ethirds__between,axiom,
( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))))
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))))
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Eextreal_2Ehalf__cancel,axiom,
c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Eextreal_2Ethird__cancel,axiom,
c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Eextreal_2Efourth__cancel,axiom,
c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Eextreal_2Equotient__normal,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(V0n_2E0),c_2Eextreal_2Eextreal__of__num_2E1(V1m_2E0)) = c_2Eextreal_2ENormal_2E1(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Ereal_2Ereal__of__num_2E1(V1m_2E0))) ) ).
tff(thm_2Eextreal_2Eext__mono__increasing__suc,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( p(c_2Eextreal_2Eext__mono__increasing_2E1(V0f_2E0))
<=> ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V1n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)))) ) ).
tff(thm_2Eextreal_2Eext__mono__decreasing__suc,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( p(c_2Eextreal_2Eext__mono__decreasing_2E1(V0f_2E0))
<=> ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V1n_2E0))) ) ).
tff(thm_2Eextreal_2EEXTREAL__ARCH,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ! [V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( V1y_2E0 != c_2Eextreal_2EPosInf_2E0 )
=> ? [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__of__num_2E1(V2n_2E0),V0x_2E0))) ) ) ).
tff(thm_2Eextreal_2ESIMP__REAL__ARCH,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))) ).
tff(thm_2Eextreal_2ESIMP__REAL__ARCH__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
? [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)),V0x_2E0)) ).
tff(thm_2Eextreal_2ESIMP__EXTREAL__ARCH,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
=> ? [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,c_2Eextreal_2Eextreal__of__num_2E1(V1n_2E0))) ) ).
tff(thm_2Eextreal_2EREAL__ARCH__POW,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
? [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V1n_2E0))) ).
tff(thm_2Eextreal_2EEXTREAL__ARCH__POW,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( V0x_2E0 != c_2Eextreal_2EPosInf_2E0 )
=> ? [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,c_2Eextreal_2Eextreal__pow_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V1n_2E0))) ) ).
tff(thm_2Eextreal_2EEXTREAL__ARCH__POW__INV,axiom,
! [V0e_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0e_2E0))
=> ? [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2ENormal_2E1(c_2Ereal_2Epow_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)))),V1n_2E0)),V0e_2E0)) ) ).
tff(thm_2Eextreal_2EREAL__LE__MUL__EPSILON,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))
& p(c_2Erealax_2Ereal__lt_2E2(V2z_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__mul_2E2(V2z_2E0,V0x_2E0),V1y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Eextreal_2Ele__epsilon,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ! [V2e_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& ( V2e_2E0 != c_2Eextreal_2EPosInf_2E0 ) )
=> p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,c_2Eextreal_2Eextreal__add_2E2(V1y_2E0,V2e_2E0))) )
=> p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Eextreal_2Ele__mul__epsilon,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ! [V2z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V2z_2E0,c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__mul_2E2(V2z_2E0,V0x_2E0),V1y_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__THM,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)] :
( ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a)) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) )
& ! [V1e_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,c_2Epred__set_2EINSERT_2E2(A_27a,V1e_2E0,V2s_2E0)) = c_2Eextreal_2Eextreal__add_2E2(app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V1e_2E0),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,c_2Epred__set_2EDELETE_2E2(A_27a,V2s_2E0,V1e_2E0))) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NOT__NEG__INF,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
& ! [V2x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V2x_2E0) != c_2Eextreal_2ENegInf_2E0 ) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,V1s_2E0) != c_2Eextreal_2ENegInf_2E0 ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NOT__POS__INF,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
& ! [V2x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V2x_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,V1s_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NOT__INFTY,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
& ! [V2x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V2x_2E0) != c_2Eextreal_2ENegInf_2E0 ) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,V1s_2E0) != c_2Eextreal_2ENegInf_2E0 ) )
& ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
& ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1s_2E0))
=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V3x_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,V1s_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__SING,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V1e_2E0: A_27a] : ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,c_2Epred__set_2EINSERT_2E2(A_27a,V1e_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a))) = app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V1e_2E0) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__POS,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
& ! [V2x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V2x_2E0))) ) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__SPOS,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
& ( V1s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
& ! [V2x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V2x_2E0))) ) )
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IF__ELIM,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)))] :
( ! [V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V1P_2E0),V2f_2E0),V4x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V2f_2E0,V4x_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
& ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V3x_2E0)) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V1P_2E0),V2f_2E0),V0s_2E0) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V2f_2E0,V0s_2E0) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__FINITE__SAME,axiom,
! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2p_2E0: A_27a] :
( ( p(c_2Ebool_2EIN_2E2(A_27a,V2p_2E0,V0s_2E0))
& ! [V3q_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3q_2E0,V0s_2E0))
=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V2p_2E0) = app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V3q_2E0) ) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0s_2E0) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0)),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V2p_2E0)) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__FINITE__CONST,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0P_2E0))
=> ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2x_2E0: tyop_2Eextreal_2Eextreal] :
( ! [V3y_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V3y_2E0) = V2x_2E0 )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0P_2E0) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27a,V0P_2E0)),V2x_2E0) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__ZERO,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)] :
( ! [V1x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,F0_2E0,V1x_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,F0_2E0,V0s_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__0,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
& ! [V2x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V2x_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,V1s_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IN__IF,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V0s_2E0),V1f_2E0),V2x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V0s_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V2x_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)] : ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0s_2E0) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V0s_2E0),V1f_2E0),V0s_2E0) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__CMUL,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)))] :
( ! [V2c_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V2c_2E0),V1f_2E0),V4x_2E0) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V2c_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V4x_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2c_2E0))
| ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V3x_2E0))) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V2c_2E0),V1f_2E0),V0s_2E0) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V2c_2E0),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0s_2E0)) ) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__CMUL2,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)))] :
( ! [V2c_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V2c_2E0),V1f_2E0),V4x_2E0) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V2c_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V4x_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
& ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V3x_2E0) != c_2Eextreal_2ENegInf_2E0 ) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V2c_2E0),V1f_2E0),V0s_2E0) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V2c_2E0),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0s_2E0)) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ! [V1f_27_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( p(c_2Epred__set_2EINJ_2E3(A_27a,A_27b,V1f_27_2E0,V0s_2E0,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V1f_27_2E0,V0s_2E0)))
=> ! [V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)] : ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27b,V2f_2E0,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V1f_27_2E0,V0s_2E0)) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Eextreal_2Eextreal,A_27b,V2f_2E0,V1f_27_2E0),V0s_2E0) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__DISJOINT__UNION,axiom,
! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
& p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_27_2E0))
& p(c_2Epred__set_2EDISJOINT_2E2(A_27a,V0s_2E0,V1s_27_2E0)) )
=> ! [V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)] : ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V2f_2E0,c_2Epred__set_2EUNION_2E2(A_27a,V0s_2E0,V1s_27_2E0)) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V2f_2E0,V0s_2E0),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V2f_2E0,V1s_27_2E0)) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__EQ__CARD,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),F0_2E0,V0s_2E0),V1x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,c_2Ebool_2EIN_2E2(A_27a,V1x_2E0,V0s_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),F0_2E0,V0s_2E0),V0s_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0)) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__INV__CARD__EQ__1,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),F0_2E0,V0s_2E0),V1x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,c_2Ebool_2EIN_2E2(A_27a,V1x_2E0,V0s_2E0),c_2Eextreal_2Eextreal__inv_2E1(c_2Eextreal_2Eextreal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0))),c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
& p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0)) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),F0_2E0,V0s_2E0),V0s_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__INTER__NONZERO,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2p_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V1f_2E0),V2p_2E0))
<=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V2p_2E0) != c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)] : ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,c_2Epred__set_2EINTER_2E2(A_27a,V0s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V1f_2E0))) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0s_2E0) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__INTER__ELIM,axiom,
! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2s_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ! [V3x_2E0: A_27a] :
( ~ p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_27_2E0))
=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V3x_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,c_2Epred__set_2EINTER_2E2(A_27a,V0s_2E0,V2s_27_2E0)) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0s_2E0) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__ZERO__DIFF,axiom,
! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2t_2E0))
=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V3x_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0s_2E0) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,c_2Epred__set_2EDIFF_2E2(A_27a,V0s_2E0,V2t_2E0)) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__MONO,axiom,
! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2f_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)] :
( ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V2f_27_2E0,V3x_2E0))) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0s_2E0),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V2f_27_2E0,V0s_2E0))) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__MONO__SET,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
& p(c_2Epred__set_2EFINITE_2E1(A_27a,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(A_27a,V1s_2E0,V2t_2E0))
& ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2t_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V3x_2E0))) ) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,V1s_2E0),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,V2t_2E0))) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__EQ,axiom,
! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2f_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)] :
( ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V3x_2E0) = app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V2f_27_2E0,V3x_2E0) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0s_2E0) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V2f_27_2E0,V0s_2E0) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__POS__MEM__LE,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
& ! [V2x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V2x_2E0))) ) )
=> ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1s_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V3x_2E0),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,V1s_2E0))) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__ADD,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2f_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V1f_2E0),V2f_27_2E0),V3x_2E0) = c_2Eextreal_2Eextreal__add_2E2(app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V2f_27_2E0,V3x_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2f_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)] : ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V1f_2E0),V2f_27_2E0),V0s_2E0) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0s_2E0),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V2f_27_2E0,V0s_2E0)) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__SUB,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2f_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V1f_2E0),V2f_27_2E0),V5x_2E0) = c_2Eextreal_2Eextreal__sub_2E2(app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V5x_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V2f_27_2E0,V5x_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2f_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)] :
( ( ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V2f_27_2E0,V3x_2E0) != c_2Eextreal_2ENegInf_2E0 ) )
| ! [V4x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V0s_2E0))
=> ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V2f_27_2E0,V4x_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V1f_2E0),V2f_27_2E0),V0s_2E0) = c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0s_2E0),c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V2f_27_2E0,V0s_2E0)) ) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__EXTREAL__SUM__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)))] :
( ! [V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)),V4x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),F1_2E0,V2f_2E0),V4x_2E0) = app_2E2(A_27b,tyop_2Eextreal_2Eextreal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal),V2f_2E0,c_2Epair_2EFST_2E1(A_27a,A_27b,V4x_2E0)),c_2Epair_2ESND_2E1(A_27a,A_27b,V4x_2E0)) )
=> ( ! [V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)),V1s_27_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V2f_2E0),V1s_27_2E0),V3x_2E0) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal),V2f_2E0,V3x_2E0),V1s_27_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_27_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal))] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
& p(c_2Epred__set_2EFINITE_2E1(A_27b,V1s_27_2E0)) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V2f_2E0),V1s_27_2E0),V0s_2E0) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),F1_2E0,V2f_2E0),c_2Epred__set_2ECROSS_2E2(A_27a,A_27b,V0s_2E0,V1s_27_2E0)) ) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NORMAL,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),F0_2E0,V0f_2E0),V2x_2E0) = c_2Eextreal_2ENormal_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),F0_2E0,V0f_2E0),V1s_2E0) = c_2Eextreal_2ENormal_2E1(c_2Ereal__sigma_2EREAL__SUM__IMAGE_2E2(A_27a,V0f_2E0,V1s_2E0)) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IN__IF__ALT,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal))))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2z_2E0: tyop_2Eextreal_2Eextreal,V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal))),F0_2E0,V0s_2E0),V1f_2E0),V2z_2E0),V3x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eextreal_2Eextreal,c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V1f_2E0,V3x_2E0),V2z_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2z_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,V1f_2E0,V0s_2E0) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(A_27a,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal))),F0_2E0,V0s_2E0),V1f_2E0),V2z_2E0),V0s_2E0) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__CROSS__SYM,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),V5y_2E0: A_27b,V6x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F1_2E0,V0f_2E0),V5y_2E0),V6x_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27b,V6x_2E0,V5y_2E0)) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),V3x_2E0: A_27a,V4y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Eextreal_2Eextreal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)),F0_2E0,V0f_2E0),V3x_2E0),V4y_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27b,V3x_2E0,V4y_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),V1s1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2s2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s1_2E0))
& p(c_2Epred__set_2EFINITE_2E1(A_27b,V2s2_2E0)) )
=> ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Eextreal_2Eextreal)),F0_2E0,V0f_2E0)),c_2Epred__set_2ECROSS_2E2(A_27a,A_27b,V1s1_2E0,V2s2_2E0)) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Epair_2Eprod(A_27b,A_27a),c_2Epair_2EUNCURRY_2E1(A_27b,A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F1_2E0,V0f_2E0)),c_2Epred__set_2ECROSS_2E2(A_27b,A_27a,V2s2_2E0,V1s1_2E0)) ) ) ) ) ).
tff(thm_2Eextreal_2EEXTREAL__SUM__IMAGE__COUNT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Enum_2Enum,V0f_2E0,c_2Epred__set_2Ecount_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Eextreal_2Eextreal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) )
& ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Enum_2Enum,V0f_2E0,c_2Epred__set_2Ecount_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) )
& ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Enum_2Enum,V0f_2E0,c_2Epred__set_2Ecount_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
& ( c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Enum_2Enum,V0f_2E0,c_2Epred__set_2Ecount_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ) ).
tff(thm_2Eextreal_2Ele__sup__imp,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V1x_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V1x_2E0,c_2Eextreal_2Eextreal__sup_2E1(V0p_2E0))) ) ).
tff(thm_2Eextreal_2Esup__le,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sup_2E1(V0p_2E0),V1x_2E0))
<=> ! [V2y_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V2y_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V2y_2E0,V1x_2E0)) ) ) ).
tff(thm_2Eextreal_2Ele__sup,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(V1x_2E0,c_2Eextreal_2Eextreal__sup_2E1(V0p_2E0)))
<=> ! [V2y_2E0: tyop_2Eextreal_2Eextreal] :
( ! [V3z_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V3z_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V3z_2E0,V2y_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(V1x_2E0,V2y_2E0)) ) ) ).
tff(thm_2Eextreal_2Esup__eq,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__sup_2E1(V0p_2E0) = V1x_2E0 )
<=> ( ! [V2y_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V2y_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V2y_2E0,V1x_2E0)) )
& ! [V3y_2E0: tyop_2Eextreal_2Eextreal] :
( ! [V4z_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V4z_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V4z_2E0,V3y_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(V1x_2E0,V3y_2E0)) ) ) ) ).
tff(thm_2Eextreal_2Esup__const,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1y_2E0))
<=> ( V1y_2E0 = V0x_2E0 ) )
=> ! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__sup_2E1(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0)) = V0x_2E0 ) ) ).
tff(thm_2Eextreal_2Esup__const__alt,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ? [V2x_2E0: tyop_2Eextreal_2Eextreal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0))
& ! [V3x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V3x_2E0))
=> ( V3x_2E0 = V1z_2E0 ) ) )
=> ( c_2Eextreal_2Eextreal__sup_2E1(V0p_2E0) = V1z_2E0 ) ) ).
tff(thm_2Eextreal_2Esup__const__over__set,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal))] :
( ! [V1k_2E0: tyop_2Eextreal_2Eextreal,V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),F0_2E0,V1k_2E0),V2x_2E0) = V1k_2E0 )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1k_2E0: tyop_2Eextreal_2Eextreal] :
( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
=> ( c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),F0_2E0,V1k_2E0),V0s_2E0)) = V1k_2E0 ) ) ) ).
tff(thm_2Eextreal_2Esup__num,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] :
( ! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,F0_2E0,V0x_2E0))
<=> ? [V1n_2E0: tyop_2Enum_2Enum] : ( V0x_2E0 = c_2Eextreal_2Eextreal__of__num_2E1(V1n_2E0) ) )
=> ( c_2Eextreal_2Eextreal__sup_2E1(F0_2E0) = c_2Eextreal_2EPosInf_2E0 ) ) ).
tff(thm_2Eextreal_2Esup__le__sup__imp,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1q_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] :
( ! [V2x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0))
=> ? [V3y_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V1q_2E0,V3y_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V2x_2E0,V3y_2E0)) ) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sup_2E1(V0p_2E0),c_2Eextreal_2Eextreal__sup_2E1(V1q_2E0))) ) ).
tff(thm_2Eextreal_2Esup__mono,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1q_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0p_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1q_2E0,V2n_2E0)))
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0p_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))),c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1q_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))))) ) ).
tff(thm_2Eextreal_2Esup__suc,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F0_2E0,V0f_2E0),V3n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V1m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0))) )
=> ( c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F0_2E0,V0f_2E0),c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) = c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) ) ) ) ).
tff(thm_2Eextreal_2Esup__seq,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F0_2E0,V0f_2E0),V2n_2E0) = c_2Eextreal_2ENormal_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eextreal_2Emono__increasing_2E1(V0f_2E0))
=> ( p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V1l_2E0))
<=> ( c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F0_2E0,V0f_2E0),c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) = c_2Eextreal_2ENormal_2E1(V1l_2E0) ) ) ) ) ).
tff(thm_2Eextreal_2Esup__lt__infty,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] :
( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__sup_2E1(V0p_2E0),c_2Eextreal_2EPosInf_2E0))
=> ! [V1x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V1x_2E0))
=> p(c_2Eextreal_2Eextreal__lt_2E2(V1x_2E0,c_2Eextreal_2EPosInf_2E0)) ) ) ).
tff(thm_2Eextreal_2Esup__max,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V1z_2E0))
& ! [V2x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V2x_2E0,V1z_2E0)) ) )
=> ( c_2Eextreal_2Eextreal__sup_2E1(V0p_2E0) = V1z_2E0 ) ) ).
tff(thm_2Eextreal_2Esup__add__mono,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V6n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0,V0f_2E0),V1g_2E0),V6n_2E0) = c_2Eextreal_2Eextreal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V6n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1g_2E0,V6n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0)))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Enum_2ESUC_2E1(V3n_2E0))))
& ! [V4n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1g_2E0,V4n_2E0)))
& ! [V5n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1g_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1g_2E0,c_2Enum_2ESUC_2E1(V5n_2E0)))) )
=> ( c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0,V0f_2E0),V1g_2E0),c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))),c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1g_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum)))) ) ) ) ).
tff(thm_2Eextreal_2Esup__sum__mono,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),V8i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F1_2E0,V0f_2E0),V8i_2E0) = c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V0f_2E0,V8i_2E0),c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),V6n_2E0: tyop_2Enum_2Enum,V7i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F2_2E0,V0f_2E0),V6n_2E0),V7i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V0f_2E0,V7i_2E0),V6n_2E0) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V6n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0,V0f_2E0),V1s_2E0),V6n_2E0) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F2_2E0,V0f_2E0),V6n_2E0),V1s_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V1s_2E0))
& ! [V2i_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2i_2E0,V1s_2E0))
=> ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V0f_2E0,V2i_2E0),V3n_2E0))) )
& ! [V4i_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4i_2E0,V1s_2E0))
=> ! [V5n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V0f_2E0,V4i_2E0),V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V0f_2E0,V4i_2E0),c_2Enum_2ESUC_2E1(V5n_2E0)))) ) )
=> ( c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0,V0f_2E0),V1s_2E0),c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F1_2E0,V0f_2E0),V1s_2E0) ) ) ) ) ) ).
tff(thm_2Eextreal_2Esup__le__mono,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Enum_2ESUC_2E1(V2n_2E0))))
& p(c_2Eextreal_2Eextreal__lt_2E2(V1z_2E0,c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))))) )
=> ? [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(V1z_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V3n_2E0))) ) ).
tff(thm_2Eextreal_2Esup__cmul,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2n_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V1c_2E0),V0f_2E0),V2n_2E0) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V1c_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V1c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1c_2E0))
=> ( c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V1c_2E0),V0f_2E0),c_2Epred__set_2EUNIV_2E0(A_27a))) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V1c_2E0),c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Epred__set_2EUNIV_2E0(A_27a)))) ) ) ) ).
tff(thm_2Eextreal_2Esup__lt,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ? [V2x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0P_2E0,V2x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,V2x_2E0)) )
<=> p(c_2Eextreal_2Eextreal__lt_2E2(V1y_2E0,c_2Eextreal_2Eextreal__sup_2E1(V0P_2E0))) ) ).
tff(thm_2Eextreal_2Esup__lt__epsilon,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1e_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))
& ? [V2x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0P_2E0,V2x_2E0))
& ( V2x_2E0 != c_2Eextreal_2ENegInf_2E0 ) )
& ( c_2Eextreal_2Eextreal__sup_2E1(V0P_2E0) != c_2Eextreal_2EPosInf_2E0 ) )
=> ? [V3x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__sup_2E1(V0P_2E0),c_2Eextreal_2Eextreal__add_2E2(V3x_2E0,V1e_2E0))) ) ) ).
tff(thm_2Eextreal_2Einf__le__imp,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V1x_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__inf_2E1(V0p_2E0),V1x_2E0)) ) ).
tff(thm_2Eextreal_2Ele__inf,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(V1x_2E0,c_2Eextreal_2Eextreal__inf_2E1(V0p_2E0)))
<=> ! [V2y_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V2y_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V1x_2E0,V2y_2E0)) ) ) ).
tff(thm_2Eextreal_2Einf__le,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__inf_2E1(V0p_2E0),V1x_2E0))
<=> ! [V2y_2E0: tyop_2Eextreal_2Eextreal] :
( ! [V3z_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V3z_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V2y_2E0,V3z_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(V2y_2E0,V1x_2E0)) ) ) ).
tff(thm_2Eextreal_2Einf__eq,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__inf_2E1(V0p_2E0) = V1x_2E0 )
<=> ( ! [V2y_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V2y_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V1x_2E0,V2y_2E0)) )
& ! [V3y_2E0: tyop_2Eextreal_2Eextreal] :
( ! [V4z_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V4z_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V3y_2E0,V4z_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(V3y_2E0,V1x_2E0)) ) ) ) ).
tff(thm_2Eextreal_2Einf__const,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1y_2E0))
<=> ( V1y_2E0 = V0x_2E0 ) )
=> ! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__inf_2E1(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0)) = V0x_2E0 ) ) ).
tff(thm_2Eextreal_2Einf__const__alt,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1z_2E0: tyop_2Eextreal_2Eextreal] :
( ( ? [V2x_2E0: tyop_2Eextreal_2Eextreal] : p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0))
& ! [V3x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V3x_2E0))
=> ( V3x_2E0 = V1z_2E0 ) ) )
=> ( c_2Eextreal_2Eextreal__inf_2E1(V0p_2E0) = V1z_2E0 ) ) ).
tff(thm_2Eextreal_2Einf__const__over__set,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal))] :
( ! [V1k_2E0: tyop_2Eextreal_2Eextreal,V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),F0_2E0,V1k_2E0),V2x_2E0) = V1k_2E0 )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1k_2E0: tyop_2Eextreal_2Eextreal] :
( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
=> ( c_2Eextreal_2Eextreal__inf_2E1(c_2Epred__set_2EIMAGE_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),F0_2E0,V1k_2E0),V0s_2E0)) = V1k_2E0 ) ) ) ).
tff(thm_2Eextreal_2Einf__suc,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F0_2E0,V0f_2E0),V3n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V1m_2E0))) )
=> ( c_2Eextreal_2Eextreal__inf_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F0_2E0,V0f_2E0),c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) = c_2Eextreal_2Eextreal__inf_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) ) ) ) ).
tff(thm_2Eextreal_2Einf__seq,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F0_2E0,V0f_2E0),V2n_2E0) = c_2Eextreal_2ENormal_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eextreal_2Emono__decreasing_2E1(V0f_2E0))
=> ( p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V1l_2E0))
<=> ( c_2Eextreal_2Eextreal__inf_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F0_2E0,V0f_2E0),c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) = c_2Eextreal_2ENormal_2E1(V1l_2E0) ) ) ) ) ).
tff(thm_2Eextreal_2Einf__lt__infty,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool)] :
( p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2ENegInf_2E0,c_2Eextreal_2Eextreal__inf_2E1(V0p_2E0)))
=> ! [V1x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V1x_2E0))
=> p(c_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2ENegInf_2E0,V1x_2E0)) ) ) ).
tff(thm_2Eextreal_2Einf__min,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),V1z_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V1z_2E0))
& ! [V2x_2E0: tyop_2Eextreal_2Eextreal] :
( p(app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0))
=> p(c_2Eextreal_2Eextreal__le_2E2(V1z_2E0,V2x_2E0)) ) )
=> ( c_2Eextreal_2Eextreal__inf_2E1(V0p_2E0) = V1z_2E0 ) ) ).
tff(thm_2Eextreal_2Einf__cminus,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V2n_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V1c_2E0),V0f_2E0),V2n_2E0) = c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2ENormal_2E1(V1c_2E0),app_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),V1c_2E0: tyop_2Erealax_2Ereal] : ( c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2ENormal_2E1(V1c_2E0),c_2Eextreal_2Eextreal__inf_2E1(c_2Epred__set_2EIMAGE_2E2(A_27a,tyop_2Eextreal_2Eextreal,V0f_2E0,c_2Epred__set_2EUNIV_2E0(A_27a)))) = c_2Eextreal_2Eextreal__sup_2E1(c_2Epred__set_2EIMAGE_2E2(A_27a,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(A_27a,tyop_2Eextreal_2Eextreal)),F0_2E0,V1c_2E0),V0f_2E0),c_2Epred__set_2EUNIV_2E0(A_27a))) ) ) ).
tff(thm_2Eextreal_2Eext__suminf__add,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0,V0f_2E0),V1g_2E0),V3n_2E0) = c_2Eextreal_2Eextreal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1g_2E0,V3n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0)))
& p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1g_2E0,V2n_2E0))) )
=> ( c_2Eextreal_2Eext__suminf_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0,V0f_2E0),V1g_2E0)) = c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2Eext__suminf_2E1(V0f_2E0),c_2Eextreal_2Eext__suminf_2E1(V1g_2E0)) ) ) ) ).
tff(thm_2Eextreal_2Eext__suminf__cmul,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)))] :
( ! [V1c_2E0: tyop_2Eextreal_2Eextreal,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0,V1c_2E0),V0f_2E0),V3n_2E0) = c_2Eextreal_2Eextreal__mul_2E2(V1c_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V3n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1c_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V1c_2E0))
& ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0))) )
=> ( c_2Eextreal_2Eext__suminf_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0,V1c_2E0),V0f_2E0)) = c_2Eextreal_2Eextreal__mul_2E2(V1c_2E0,c_2Eextreal_2Eext__suminf_2E1(V0f_2E0)) ) ) ) ).
tff(thm_2Eextreal_2Eext__suminf__cmul__alt,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0,V1c_2E0),V0f_2E0),V4n_2E0) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V1c_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V4n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1c_2E0))
& ( ! [V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0) != c_2Eextreal_2ENegInf_2E0 )
| ! [V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V3n_2E0) != c_2Eextreal_2EPosInf_2E0 ) ) )
=> ( c_2Eextreal_2Eext__suminf_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0,V1c_2E0),V0f_2E0)) = c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2ENormal_2E1(V1c_2E0),c_2Eextreal_2Eext__suminf_2E1(V0f_2E0)) ) ) ) ).
tff(thm_2Eextreal_2Eext__suminf__lt__infty,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( ( ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V1n_2E0)))
& ( c_2Eextreal_2Eext__suminf_2E1(V0f_2E0) != c_2Eextreal_2EPosInf_2E0 ) )
=> ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__lt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0),c_2Eextreal_2EPosInf_2E0)) ) ).
tff(thm_2Eextreal_2Eext__suminf__suminf,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal))] :
( ! [V0r_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F1_2E0,V0r_2E0),V3n_2E0) = c_2Eextreal_2ENormal_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0r_2E0,V3n_2E0)) )
=> ( ! [V0r_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F0_2E0,V0r_2E0),V2n_2E0) = c_2Eextreal_2ENormal_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0r_2E0,V2n_2E0)) )
=> ! [V0r_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,V0r_2E0,V1n_2E0)))
& ( c_2Eextreal_2Eext__suminf_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F0_2E0,V0r_2E0)) != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eext__suminf_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),F1_2E0,V0r_2E0)) = c_2Eextreal_2ENormal_2E1(c_2Eseq_2Esuminf_2E1(V0r_2E0)) ) ) ) ) ).
tff(thm_2Eextreal_2Eext__suminf__mono,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1g_2E0,V2n_2E0) != c_2Eextreal_2ENegInf_2E0 )
& p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1g_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0))) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eext__suminf_2E1(V1g_2E0),c_2Eextreal_2Eext__suminf_2E1(V0f_2E0))) ) ).
tff(thm_2Eextreal_2Eext__suminf__sub,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0,V0f_2E0),V1g_2E0),V3i_2E0) = c_2Eextreal_2Eextreal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V3i_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1g_2E0,V3i_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1g_2E0,V2n_2E0)))
& p(c_2Eextreal_2Eextreal__le_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V1g_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0))) )
& ( c_2Eextreal_2Eext__suminf_2E1(V0f_2E0) != c_2Eextreal_2EPosInf_2E0 ) )
=> ( c_2Eextreal_2Eext__suminf_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal)),F0_2E0,V0f_2E0),V1g_2E0)) = c_2Eextreal_2Eextreal__sub_2E2(c_2Eextreal_2Eext__suminf_2E1(V0f_2E0),c_2Eextreal_2Eext__suminf_2E1(V1g_2E0)) ) ) ) ).
tff(thm_2Eextreal_2Eext__suminf__sum,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal),V1n_2E0: tyop_2Enum_2Enum] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V2n_2E0)))
& ! [V3m_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V3m_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Eextreal_2Eextreal,V0f_2E0,V3m_2E0) = c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
=> ( c_2Eextreal_2Eext__suminf_2E1(V0f_2E0) = c_2Eextreal_2EEXTREAL__SUM__IMAGE_2E2(tyop_2Enum_2Enum,V0f_2E0,c_2Epred__set_2Ecount_2E1(V1n_2E0)) ) ) ).
tff(thm_2Eextreal_2Emin__le,axiom,
! [V0z_2E0: tyop_2Eextreal_2Eextreal,V1x_2E0: tyop_2Eextreal_2Eextreal,V2y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__min_2E2(V1x_2E0,V2y_2E0),V0z_2E0))
<=> ( p(c_2Eextreal_2Eextreal__le_2E2(V1x_2E0,V0z_2E0))
| p(c_2Eextreal_2Eextreal__le_2E2(V2y_2E0,V0z_2E0)) ) ) ).
tff(thm_2Eextreal_2Emin__le1,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__min_2E2(V0x_2E0,V1y_2E0),V0x_2E0)) ).
tff(thm_2Eextreal_2Emin__le2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__min_2E2(V0x_2E0,V1y_2E0),V1y_2E0)) ).
tff(thm_2Eextreal_2Ele__min,axiom,
! [V0z_2E0: tyop_2Eextreal_2Eextreal,V1x_2E0: tyop_2Eextreal_2Eextreal,V2y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(V0z_2E0,c_2Eextreal_2Eextreal__min_2E2(V1x_2E0,V2y_2E0)))
<=> ( p(c_2Eextreal_2Eextreal__le_2E2(V0z_2E0,V1x_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V0z_2E0,V2y_2E0)) ) ) ).
tff(thm_2Eextreal_2Emin__le2__imp,axiom,
! [V0x1_2E0: tyop_2Eextreal_2Eextreal,V1x2_2E0: tyop_2Eextreal_2Eextreal,V2y1_2E0: tyop_2Eextreal_2Eextreal,V3y2_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(V0x1_2E0,V2y1_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V1x2_2E0,V3y2_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__min_2E2(V0x1_2E0,V1x2_2E0),c_2Eextreal_2Eextreal__min_2E2(V2y1_2E0,V3y2_2E0))) ) ).
tff(thm_2Eextreal_2Emin__refl,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__min_2E2(V0x_2E0,V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextreal_2Emin__comm,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__min_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__min_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Eextreal_2Emin__infty,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__min_2E2(V0x_2E0,c_2Eextreal_2EPosInf_2E0) = V0x_2E0 )
& ( c_2Eextreal_2Eextreal__min_2E2(c_2Eextreal_2EPosInf_2E0,V0x_2E0) = V0x_2E0 )
& ( c_2Eextreal_2Eextreal__min_2E2(c_2Eextreal_2ENegInf_2E0,V0x_2E0) = c_2Eextreal_2ENegInf_2E0 )
& ( c_2Eextreal_2Eextreal__min_2E2(V0x_2E0,c_2Eextreal_2ENegInf_2E0) = c_2Eextreal_2ENegInf_2E0 ) ) ).
tff(thm_2Eextreal_2Ele__max,axiom,
! [V0z_2E0: tyop_2Eextreal_2Eextreal,V1x_2E0: tyop_2Eextreal_2Eextreal,V2y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(V0z_2E0,c_2Eextreal_2Eextreal__max_2E2(V1x_2E0,V2y_2E0)))
<=> ( p(c_2Eextreal_2Eextreal__le_2E2(V0z_2E0,V1x_2E0))
| p(c_2Eextreal_2Eextreal__le_2E2(V0z_2E0,V2y_2E0)) ) ) ).
tff(thm_2Eextreal_2Ele__max1,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : p(c_2Eextreal_2Eextreal__le_2E2(V0x_2E0,c_2Eextreal_2Eextreal__max_2E2(V0x_2E0,V1y_2E0))) ).
tff(thm_2Eextreal_2Ele__max2,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : p(c_2Eextreal_2Eextreal__le_2E2(V1y_2E0,c_2Eextreal_2Eextreal__max_2E2(V0x_2E0,V1y_2E0))) ).
tff(thm_2Eextreal_2Emax__le,axiom,
! [V0z_2E0: tyop_2Eextreal_2Eextreal,V1x_2E0: tyop_2Eextreal_2Eextreal,V2y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__max_2E2(V1x_2E0,V2y_2E0),V0z_2E0))
<=> ( p(c_2Eextreal_2Eextreal__le_2E2(V1x_2E0,V0z_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V2y_2E0,V0z_2E0)) ) ) ).
tff(thm_2Eextreal_2Emax__le2__imp,axiom,
! [V0x1_2E0: tyop_2Eextreal_2Eextreal,V1x2_2E0: tyop_2Eextreal_2Eextreal,V2y1_2E0: tyop_2Eextreal_2Eextreal,V3y2_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(V0x1_2E0,V2y1_2E0))
& p(c_2Eextreal_2Eextreal__le_2E2(V1x2_2E0,V3y2_2E0)) )
=> p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__max_2E2(V0x1_2E0,V1x2_2E0),c_2Eextreal_2Eextreal__max_2E2(V2y1_2E0,V3y2_2E0))) ) ).
tff(thm_2Eextreal_2Emax__refl,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__max_2E2(V0x_2E0,V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextreal_2Emax__comm,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__max_2E2(V0x_2E0,V1y_2E0) = c_2Eextreal_2Eextreal__max_2E2(V1y_2E0,V0x_2E0) ) ).
tff(thm_2Eextreal_2Emax__infty,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( c_2Eextreal_2Eextreal__max_2E2(V0x_2E0,c_2Eextreal_2EPosInf_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__max_2E2(c_2Eextreal_2EPosInf_2E0,V0x_2E0) = c_2Eextreal_2EPosInf_2E0 )
& ( c_2Eextreal_2Eextreal__max_2E2(c_2Eextreal_2ENegInf_2E0,V0x_2E0) = V0x_2E0 )
& ( c_2Eextreal_2Eextreal__max_2E2(V0x_2E0,c_2Eextreal_2ENegInf_2E0) = V0x_2E0 ) ) ).
tff(thm_2Eextreal_2EQ__not__infty,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V0x_2E0,c_2Eextreal_2EQ__set_2E0))
=> ? [V1y_2E0: tyop_2Erealax_2Ereal] : ( V0x_2E0 = c_2Eextreal_2ENormal_2E1(V1y_2E0) ) ) ).
tff(thm_2Eextreal_2EQ__COUNTABLE,axiom,
p(c_2Epred__set_2Ecountable_2E1(tyop_2Eextreal_2Eextreal,c_2Eextreal_2EQ__set_2E0)) ).
tff(thm_2Eextreal_2ENUM__IN__Q,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__of__num_2E1(V0n_2E0),c_2Eextreal_2EQ__set_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2Eextreal__of__num_2E1(V0n_2E0)),c_2Eextreal_2EQ__set_2E0)) ) ).
tff(thm_2Eextreal_2EQ__INFINITE,axiom,
~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Eextreal_2Eextreal,c_2Eextreal_2EQ__set_2E0)) ).
tff(thm_2Eextreal_2EOPP__IN__Q,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V0x_2E0,c_2Eextreal_2EQ__set_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__ainv_2E1(V0x_2E0),c_2Eextreal_2EQ__set_2E0)) ) ).
tff(thm_2Eextreal_2EINV__IN__Q,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V0x_2E0,c_2Eextreal_2EQ__set_2E0))
& ( V0x_2E0 != c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__div_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0),c_2Eextreal_2EQ__set_2E0)) ) ).
tff(thm_2Eextreal_2ECMUL__IN__Q,axiom,
! [V0z_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V1x_2E0,c_2Eextreal_2EQ__set_2E0))
=> ( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__of__num_2E1(V0z_2E0),V1x_2E0),c_2Eextreal_2EQ__set_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__mul_2E2(c_2Eextreal_2Eextreal__ainv_2E1(c_2Eextreal_2Eextreal__of__num_2E1(V0z_2E0)),V1x_2E0),c_2Eextreal_2EQ__set_2E0)) ) ) ).
tff(thm_2Eextreal_2EADD__IN__Q,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V0x_2E0,c_2Eextreal_2EQ__set_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V1y_2E0,c_2Eextreal_2EQ__set_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__add_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2EQ__set_2E0)) ) ).
tff(thm_2Eextreal_2ESUB__IN__Q,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V0x_2E0,c_2Eextreal_2EQ__set_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V1y_2E0,c_2Eextreal_2EQ__set_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__sub_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2EQ__set_2E0)) ) ).
tff(thm_2Eextreal_2EMUL__IN__Q,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V0x_2E0,c_2Eextreal_2EQ__set_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V1y_2E0,c_2Eextreal_2EQ__set_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__mul_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2EQ__set_2E0)) ) ).
tff(thm_2Eextreal_2EDIV__IN__Q,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V0x_2E0,c_2Eextreal_2EQ__set_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V1y_2E0,c_2Eextreal_2EQ__set_2E0))
& ( V1y_2E0 != c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,c_2Eextreal_2Eextreal__div_2E2(V0x_2E0,V1y_2E0),c_2Eextreal_2EQ__set_2E0)) ) ).
tff(thm_2Eextreal_2Erat__not__infty,axiom,
! [V0r_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V0r_2E0,c_2Eextreal_2EQ__set_2E0))
=> ( ( V0r_2E0 != c_2Eextreal_2ENegInf_2E0 )
& ( V0r_2E0 != c_2Eextreal_2EPosInf_2E0 ) ) ) ).
tff(thm_2Eextreal_2ECEILING__LBOUND,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2ENormal_2E1(V0x_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Eextreal_2Eceiling_2E1(c_2Eextreal_2ENormal_2E1(V0x_2E0))))) ).
tff(thm_2Eextreal_2ECEILING__UBOUND,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_2Eextreal_2Eextreal__lt_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Eextreal_2Eceiling_2E1(c_2Eextreal_2ENormal_2E1(V0x_2E0))),c_2Eextreal_2Eextreal__add_2E2(c_2Eextreal_2ENormal_2E1(V0x_2E0),c_2Eextreal_2Eextreal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) ) ).
tff(thm_2Eextreal_2EQ__DENSE__IN__R__LEMMA,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( ( p(c_2Eextreal_2Eextreal__le_2E2(c_2Eextreal_2Eextreal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0)) )
=> ? [V2r_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V2r_2E0,c_2Eextreal_2EQ__set_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V2r_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V2r_2E0,V1y_2E0)) ) ) ).
tff(thm_2Eextreal_2EQ__DENSE__IN__R,axiom,
! [V0x_2E0: tyop_2Eextreal_2Eextreal,V1y_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V1y_2E0))
=> ? [V2r_2E0: tyop_2Eextreal_2Eextreal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Eextreal_2Eextreal,V2r_2E0,c_2Eextreal_2EQ__set_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V0x_2E0,V2r_2E0))
& p(c_2Eextreal_2Eextreal__lt_2E2(V2r_2E0,V1y_2E0)) ) ) ).
tff(thm_2Eextreal_2ECOUNTABLE__ENUM__Q,axiom,
! [A_27a: $tType,V0c_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2Ecountable_2E1(A_27a,V0c_2E0))
<=> ( ( V0c_2E0 = c_2Epred__set_2EEMPTY_2E0(A_27a) )
| ? [V1f_2E0: tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,A_27a)] : ( V0c_2E0 = c_2Epred__set_2EIMAGE_2E2(tyop_2Eextreal_2Eextreal,A_27a,V1f_2E0,c_2Eextreal_2EQ__set_2E0) ) ) ) ).
tff(thm_2Eextreal_2ECROSS__COUNTABLE__UNIV,axiom,
p(c_2Epred__set_2Ecountable_2E1(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Epred__set_2ECROSS_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum),c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum)))) ).
tff(thm_2Eextreal_2ECROSS__COUNTABLE__LEMMA1,axiom,
! [A_27a: $tType,A_27b: $tType,V0t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2Ecountable_2E1(A_27a,V1s_2E0))
& p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
& p(c_2Epred__set_2Ecountable_2E1(A_27b,V0t_2E0)) )
=> p(c_2Epred__set_2Ecountable_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epred__set_2ECROSS_2E2(A_27a,A_27b,V1s_2E0,V0t_2E0))) ) ).
tff(thm_2Eextreal_2ECROSS__COUNTABLE__LEMMA2,axiom,
! [A_27a: $tType,A_27b: $tType,V0t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2Ecountable_2E1(A_27a,V1s_2E0))
& p(c_2Epred__set_2Ecountable_2E1(A_27b,V0t_2E0))
& p(c_2Epred__set_2EFINITE_2E1(A_27b,V0t_2E0)) )
=> p(c_2Epred__set_2Ecountable_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epred__set_2ECROSS_2E2(A_27a,A_27b,V1s_2E0,V0t_2E0))) ) ).
tff(thm_2Eextreal_2ECROSS__COUNTABLE,axiom,
! [A_27a: $tType,A_27b: $tType,V0t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2Ecountable_2E1(A_27a,V1s_2E0))
& p(c_2Epred__set_2Ecountable_2E1(A_27b,V0t_2E0)) )
=> p(c_2Epred__set_2Ecountable_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epred__set_2ECROSS_2E2(A_27a,A_27b,V1s_2E0,V0t_2E0))) ) ).
tff(thm_2Eextreal_2ECOUNTABLE__RATIONAL__INTERVALS,axiom,
p(c_2Epred__set_2Ecountable_2E1(tyop_2Emin_2Efun(tyop_2Eextreal_2Eextreal,tyop_2Emin_2Ebool),c_2Eextreal_2Erational__intervals_2E0)) ).
%------------------------------------------------------------------------------