ITP001 Axioms: ITP143+5.ax
%------------------------------------------------------------------------------
% File : ITP143+5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : extreal+2.ax [Gau20]
% : HL4143+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 465 ( 65 unt; 0 def)
% Number of atoms : 2213 ( 518 equ)
% Maximal formula atoms : 20 ( 4 avg)
% Number of connectives : 1949 ( 201 ~; 24 |; 367 &)
% ( 87 <=>;1270 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 173 ( 173 usr; 82 con; 0-4 aty)
% Number of variables : 1079 (1056 !; 23 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(ne_ty_2Eextreal_2Eextreal,axiom,
ne(ty_2Eextreal_2Eextreal) ).
fof(mem_c_2Eextreal_2EEXTREAL__SUM__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(arr(A_27a,bool),ty_2Eextreal_2Eextreal))) ) ).
fof(mem_c_2Eextreal_2ENegInf,axiom,
mem(c_2Eextreal_2ENegInf,ty_2Eextreal_2Eextreal) ).
fof(mem_c_2Eextreal_2ENormal,axiom,
mem(c_2Eextreal_2ENormal,arr(ty_2Erealax_2Ereal,ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2EPosInf,axiom,
mem(c_2Eextreal_2EPosInf,ty_2Eextreal_2Eextreal) ).
fof(mem_c_2Eextreal_2EQ__set,axiom,
mem(c_2Eextreal_2EQ__set,arr(ty_2Eextreal_2Eextreal,bool)) ).
fof(mem_c_2Eextreal_2Eceiling,axiom,
mem(c_2Eextreal_2Eceiling,arr(ty_2Eextreal_2Eextreal,ty_2Enum_2Enum)) ).
fof(mem_c_2Eextreal_2Eext__mono__decreasing,axiom,
mem(c_2Eextreal_2Eext__mono__decreasing,arr(arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),bool)) ).
fof(mem_c_2Eextreal_2Eext__mono__increasing,axiom,
mem(c_2Eextreal_2Eext__mono__increasing,arr(arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),bool)) ).
fof(mem_c_2Eextreal_2Eext__suminf,axiom,
mem(c_2Eextreal_2Eext__suminf,arr(arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2Eextreal__CASE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eextreal_2Eextreal__CASE(A_27a),arr(ty_2Eextreal_2Eextreal,arr(A_27a,arr(A_27a,arr(arr(ty_2Erealax_2Ereal,A_27a),A_27a))))) ) ).
fof(mem_c_2Eextreal_2Eextreal__abs,axiom,
mem(c_2Eextreal_2Eextreal__abs,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2Eextreal__add,axiom,
mem(c_2Eextreal_2Eextreal__add,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
fof(mem_c_2Eextreal_2Eextreal__ainv,axiom,
mem(c_2Eextreal_2Eextreal__ainv,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2Eextreal__div,axiom,
mem(c_2Eextreal_2Eextreal__div,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
fof(mem_c_2Eextreal_2Eextreal__exp,axiom,
mem(c_2Eextreal_2Eextreal__exp,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2Eextreal__inf,axiom,
mem(c_2Eextreal_2Eextreal__inf,arr(arr(ty_2Eextreal_2Eextreal,bool),ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2Eextreal__inv,axiom,
mem(c_2Eextreal_2Eextreal__inv,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2Eextreal__le,axiom,
mem(c_2Eextreal_2Eextreal__le,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool))) ).
fof(mem_c_2Eextreal_2Eextreal__lg,axiom,
mem(c_2Eextreal_2Eextreal__lg,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2Eextreal__logr,axiom,
mem(c_2Eextreal_2Eextreal__logr,arr(ty_2Erealax_2Ereal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
fof(mem_c_2Eextreal_2Eextreal__lt,axiom,
mem(c_2Eextreal_2Eextreal__lt,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool))) ).
fof(mem_c_2Eextreal_2Eextreal__max,axiom,
mem(c_2Eextreal_2Eextreal__max,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
fof(mem_c_2Eextreal_2Eextreal__min,axiom,
mem(c_2Eextreal_2Eextreal__min,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
fof(mem_c_2Eextreal_2Eextreal__mul,axiom,
mem(c_2Eextreal_2Eextreal__mul,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
fof(mem_c_2Eextreal_2Eextreal__of__num,axiom,
mem(c_2Eextreal_2Eextreal__of__num,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2Eextreal__pow,axiom,
mem(c_2Eextreal_2Eextreal__pow,arr(ty_2Eextreal_2Eextreal,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))) ).
fof(mem_c_2Eextreal_2Eextreal__size,axiom,
mem(c_2Eextreal_2Eextreal__size,arr(ty_2Eextreal_2Eextreal,ty_2Enum_2Enum)) ).
fof(mem_c_2Eextreal_2Eextreal__sqrt,axiom,
mem(c_2Eextreal_2Eextreal__sqrt,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2Eextreal__sub,axiom,
mem(c_2Eextreal_2Eextreal__sub,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
fof(mem_c_2Eextreal_2Eextreal__sup,axiom,
mem(c_2Eextreal_2Eextreal__sup,arr(arr(ty_2Eextreal_2Eextreal,bool),ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2Emono__decreasing,axiom,
mem(c_2Eextreal_2Emono__decreasing,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
fof(mem_c_2Eextreal_2Emono__increasing,axiom,
mem(c_2Eextreal_2Emono__increasing,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
fof(mem_c_2Eextreal_2Eopen__interval,axiom,
mem(c_2Eextreal_2Eopen__interval,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool)))) ).
fof(mem_c_2Eextreal_2Eopen__intervals__set,axiom,
mem(c_2Eextreal_2Eopen__intervals__set,arr(arr(ty_2Eextreal_2Eextreal,bool),bool)) ).
fof(mem_c_2Eextreal_2Erational__intervals,axiom,
mem(c_2Eextreal_2Erational__intervals,arr(arr(ty_2Eextreal_2Eextreal,bool),bool)) ).
fof(mem_c_2Eextreal_2Ereal,axiom,
mem(c_2Eextreal_2Ereal,arr(ty_2Eextreal_2Eextreal,ty_2Erealax_2Ereal)) ).
fof(lameq_f3341,axiom,
! [V7a] : ap(f3341,V7a) = ap(ap(ap(c_2Eind__type_2ECONSTR(ty_2Erealax_2Ereal),ap(c_2Enum_2ESUC,ap(c_2Enum_2ESUC,c_2Enum_2E0))),V7a),k(ty_2Enum_2Enum,c_2Eind__type_2EBOTTOM(ty_2Erealax_2Ereal))) ).
fof(lameq_f3342,axiom,
! [V3a0] :
( mem(V3a0,ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal))
=> ! [V6a] : ap(f3342(V3a0),V6a) = ap(ap(c_2Emin_2E_3D(ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal)),V3a0),ap(f3341,V6a)) ) ).
fof(lameq_f3343,axiom,
! [V2_27extreal_27] :
( mem(V2_27extreal_27,arr(ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal),bool))
=> ! [V3a0] : ap(f3343(V2_27extreal_27),V3a0) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Emin_2E_3D(ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal)),V3a0),ap(ap(ap(c_2Eind__type_2ECONSTR(ty_2Erealax_2Ereal),c_2Enum_2E0),c_2Ebool_2EARB(ty_2Erealax_2Ereal)),k(ty_2Enum_2Enum,c_2Eind__type_2EBOTTOM(ty_2Erealax_2Ereal))))),ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Emin_2E_3D(ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal)),V3a0),ap(ap(ap(c_2Eind__type_2ECONSTR(ty_2Erealax_2Ereal),ap(c_2Enum_2ESUC,c_2Enum_2E0)),c_2Ebool_2EARB(ty_2Erealax_2Ereal)),k(ty_2Enum_2Enum,c_2Eind__type_2EBOTTOM(ty_2Erealax_2Ereal))))),ap(c_2Ebool_2E_3F(ty_2Erealax_2Ereal),f3342(V3a0))))),ap(V2_27extreal_27,V3a0)) ) ).
fof(lameq_f3344,axiom,
! [V1a0] :
( mem(V1a0,ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal))
=> ! [V2_27extreal_27] : ap(f3344(V1a0),V2_27extreal_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal)),f3343(V2_27extreal_27))),ap(V2_27extreal_27,V1a0)) ) ).
fof(lameq_f3345,axiom,
! [V1a0] : ap(f3345,V1a0) = ap(c_2Ebool_2E_21(arr(ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal),bool)),f3344(V1a0)) ).
fof(ax_thm_2Eextreal_2Eextreal__TY__DEF,axiom,
? [V0rep] :
( mem(V0rep,arr(ty_2Eextreal_2Eextreal,ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal)))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal),ty_2Eextreal_2Eextreal),f3345),V0rep)) ) ).
fof(ax_thm_2Eextreal_2Eextreal__case__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0v] :
( mem(V0v,A_27a)
=> ! [V1v1] :
( mem(V1v1,A_27a)
=> ! [V2f] :
( mem(V2f,arr(ty_2Erealax_2Ereal,A_27a))
=> ap(ap(ap(ap(c_2Eextreal_2Eextreal__CASE(A_27a),c_2Eextreal_2ENegInf),V0v),V1v1),V2f) = V0v ) ) )
& ! [V3v] :
( mem(V3v,A_27a)
=> ! [V4v1] :
( mem(V4v1,A_27a)
=> ! [V5f] :
( mem(V5f,arr(ty_2Erealax_2Ereal,A_27a))
=> ap(ap(ap(ap(c_2Eextreal_2Eextreal__CASE(A_27a),c_2Eextreal_2EPosInf),V3v),V4v1),V5f) = V4v1 ) ) )
& ! [V6a] :
( mem(V6a,ty_2Erealax_2Ereal)
=> ! [V7v] :
( mem(V7v,A_27a)
=> ! [V8v1] :
( mem(V8v1,A_27a)
=> ! [V9f] :
( mem(V9f,arr(ty_2Erealax_2Ereal,A_27a))
=> ap(ap(ap(ap(c_2Eextreal_2Eextreal__CASE(A_27a),ap(c_2Eextreal_2ENormal,V6a)),V7v),V8v1),V9f) = ap(V9f,V6a) ) ) ) ) ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__size__def,axiom,
( ap(c_2Eextreal_2Eextreal__size,c_2Eextreal_2ENegInf) = c_2Enum_2E0
& ap(c_2Eextreal_2Eextreal__size,c_2Eextreal_2EPosInf) = c_2Enum_2E0
& ! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ap(c_2Eextreal_2Eextreal__size,ap(c_2Eextreal_2ENormal,V0a)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ) ) ).
fof(conj_thm_2Eextreal_2Edatatype__extreal,axiom,
! [V0extreal] :
( mem(V0extreal,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,arr(arr(ty_2Erealax_2Ereal,ty_2Eextreal_2Eextreal),bool))))
=> p(ap(c_2Ebool_2EDATATYPE(bool),ap(ap(ap(V0extreal,c_2Eextreal_2ENegInf),c_2Eextreal_2EPosInf),c_2Eextreal_2ENormal))) ) ).
fof(conj_thm_2Eextreal_2Eextreal__distinct,axiom,
( c_2Eextreal_2ENegInf != c_2Eextreal_2EPosInf
& ! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> c_2Eextreal_2ENegInf != ap(c_2Eextreal_2ENormal,V0a) )
& ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> c_2Eextreal_2EPosInf != ap(c_2Eextreal_2ENormal,V1a) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__nchotomy,axiom,
! [V0ee] :
( mem(V0ee,ty_2Eextreal_2Eextreal)
=> ( V0ee = c_2Eextreal_2ENegInf
| V0ee = c_2Eextreal_2EPosInf
| ? [V1r] :
( mem(V1r,ty_2Erealax_2Ereal)
& V0ee = ap(c_2Eextreal_2ENormal,V1r) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__Axiom,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f0] :
( mem(V0f0,A_27a)
=> ! [V1f1] :
( mem(V1f1,A_27a)
=> ! [V2f2] :
( mem(V2f2,arr(ty_2Erealax_2Ereal,A_27a))
=> ? [V3fn] :
( mem(V3fn,arr(ty_2Eextreal_2Eextreal,A_27a))
& ap(V3fn,c_2Eextreal_2ENegInf) = V0f0
& ap(V3fn,c_2Eextreal_2EPosInf) = V1f1
& ! [V4a] :
( mem(V4a,ty_2Erealax_2Ereal)
=> ap(V3fn,ap(c_2Eextreal_2ENormal,V4a)) = ap(V2f2,V4a) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__induction,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,bool))
=> ( ( p(ap(V0P,c_2Eextreal_2ENegInf))
& p(ap(V0P,c_2Eextreal_2EPosInf))
& ! [V1r] :
( mem(V1r,ty_2Erealax_2Ereal)
=> p(ap(V0P,ap(c_2Eextreal_2ENormal,V1r))) ) )
=> ! [V2e] :
( mem(V2e,ty_2Eextreal_2Eextreal)
=> p(ap(V0P,V2e)) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__case__cong,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0v_27] :
( mem(V0v_27,A_27a)
=> ! [V1v1_27] :
( mem(V1v1_27,A_27a)
=> ! [V2f_27] :
( mem(V2f_27,arr(ty_2Erealax_2Ereal,A_27a))
=> ! [V3M] :
( mem(V3M,ty_2Eextreal_2Eextreal)
=> ! [V4M_27] :
( mem(V4M_27,ty_2Eextreal_2Eextreal)
=> ! [V5v] :
( mem(V5v,A_27a)
=> ! [V6v1] :
( mem(V6v1,A_27a)
=> ! [V7f] :
( mem(V7f,arr(ty_2Erealax_2Ereal,A_27a))
=> ( ( V3M = V4M_27
& ( V4M_27 = c_2Eextreal_2ENegInf
=> V5v = V0v_27 )
& ( V4M_27 = c_2Eextreal_2EPosInf
=> V6v1 = V1v1_27 )
& ! [V8a] :
( mem(V8a,ty_2Erealax_2Ereal)
=> ( V4M_27 = ap(c_2Eextreal_2ENormal,V8a)
=> ap(V7f,V8a) = ap(V2f_27,V8a) ) ) )
=> ap(ap(ap(ap(c_2Eextreal_2Eextreal__CASE(A_27a),V3M),V5v),V6v1),V7f) = ap(ap(ap(ap(c_2Eextreal_2Eextreal__CASE(A_27a),V4M_27),V0v_27),V1v1_27),V2f_27) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__case__eq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1v] :
( mem(V1v,A_27a)
=> ! [V2v1] :
( mem(V2v1,A_27a)
=> ! [V3f] :
( mem(V3f,arr(ty_2Erealax_2Ereal,A_27a))
=> ! [V4v_27] :
( mem(V4v_27,A_27a)
=> ( ap(ap(ap(ap(c_2Eextreal_2Eextreal__CASE(A_27a),V0x),V1v),V2v1),V3f) = V4v_27
<=> ( ( V0x = c_2Eextreal_2ENegInf
& V1v = V4v_27 )
| ( V0x = c_2Eextreal_2EPosInf
& V2v1 = V4v_27 )
| ? [V5r] :
( mem(V5r,ty_2Erealax_2Ereal)
& V0x = ap(c_2Eextreal_2ENormal,V5r)
& ap(V3f,V5r) = V4v_27 ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__of__num__def,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Eextreal_2Eextreal__of__num,V0n) = ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,V0n)) ) ).
fof(lameq_f3346,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1r] : ap(f3346(V0x),V1r) = ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),V0x),ap(c_2Eextreal_2ENormal,V1r)) ) ).
fof(ax_thm_2Eextreal_2Ereal__def,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(c_2Eextreal_2Ereal,V0x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),V0x),c_2Eextreal_2ENegInf)),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),V0x),c_2Eextreal_2EPosInf))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3346(V0x))) ) ).
fof(conj_thm_2Eextreal_2Ereal__normal,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Eextreal_2Ereal,ap(c_2Eextreal_2ENormal,V0x)) = V0x ) ).
fof(conj_thm_2Eextreal_2Enormal__real,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
=> ap(c_2Eextreal_2ENormal,ap(c_2Eextreal_2Ereal,V0x)) = V0x ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__add__ind,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool)))
=> ( ( ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,ap(c_2Eextreal_2ENormal,V1x)),ap(c_2Eextreal_2ENormal,V2y))) ) )
& ! [V3a] :
( mem(V3a,ty_2Eextreal_2Eextreal)
=> p(ap(ap(V0P,c_2Eextreal_2EPosInf),V3a)) )
& p(ap(ap(V0P,c_2Eextreal_2ENegInf),c_2Eextreal_2EPosInf))
& ! [V4v2] :
( mem(V4v2,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,ap(c_2Eextreal_2ENormal,V4v2)),c_2Eextreal_2EPosInf)) )
& p(ap(ap(V0P,c_2Eextreal_2ENegInf),c_2Eextreal_2ENegInf))
& ! [V5v5] :
( mem(V5v5,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,c_2Eextreal_2ENegInf),ap(c_2Eextreal_2ENormal,V5v5))) )
& ! [V6v3] :
( mem(V6v3,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,ap(c_2Eextreal_2ENormal,V6v3)),c_2Eextreal_2ENegInf)) ) )
=> ! [V7v] :
( mem(V7v,ty_2Eextreal_2Eextreal)
=> ! [V8v1] :
( mem(V8v1,ty_2Eextreal_2Eextreal)
=> p(ap(ap(V0P,V7v),V8v1)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__add__def,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2a] :
( mem(V2a,ty_2Eextreal_2Eextreal)
=> ! [V3v2] :
( mem(V3v2,ty_2Erealax_2Ereal)
=> ! [V4v5] :
( mem(V4v5,ty_2Erealax_2Ereal)
=> ! [V5v3] :
( mem(V5v3,ty_2Erealax_2Ereal)
=> ( ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2ENormal,V0x)),ap(c_2Eextreal_2ENormal,V1y)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y))
& ap(ap(c_2Eextreal_2Eextreal__add,c_2Eextreal_2EPosInf),V2a) = c_2Eextreal_2EPosInf
& ap(ap(c_2Eextreal_2Eextreal__add,c_2Eextreal_2ENegInf),c_2Eextreal_2EPosInf) = c_2Eextreal_2EPosInf
& ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2ENormal,V3v2)),c_2Eextreal_2EPosInf) = c_2Eextreal_2EPosInf
& ap(ap(c_2Eextreal_2Eextreal__add,c_2Eextreal_2ENegInf),c_2Eextreal_2ENegInf) = c_2Eextreal_2ENegInf
& ap(ap(c_2Eextreal_2Eextreal__add,c_2Eextreal_2ENegInf),ap(c_2Eextreal_2ENormal,V4v5)) = c_2Eextreal_2ENegInf
& ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2ENormal,V5v3)),c_2Eextreal_2ENegInf) = c_2Eextreal_2ENegInf ) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__sub__ind,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool)))
=> ( ( ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,ap(c_2Eextreal_2ENormal,V1x)),ap(c_2Eextreal_2ENormal,V2y))) ) )
& ! [V3a] :
( mem(V3a,ty_2Eextreal_2Eextreal)
=> p(ap(ap(V0P,c_2Eextreal_2EPosInf),V3a)) )
& p(ap(ap(V0P,c_2Eextreal_2ENegInf),c_2Eextreal_2EPosInf))
& ! [V4v2] :
( mem(V4v2,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,ap(c_2Eextreal_2ENormal,V4v2)),c_2Eextreal_2EPosInf)) )
& p(ap(ap(V0P,c_2Eextreal_2ENegInf),c_2Eextreal_2ENegInf))
& ! [V5v5] :
( mem(V5v5,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,c_2Eextreal_2ENegInf),ap(c_2Eextreal_2ENormal,V5v5))) )
& ! [V6v3] :
( mem(V6v3,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,ap(c_2Eextreal_2ENormal,V6v3)),c_2Eextreal_2ENegInf)) ) )
=> ! [V7v] :
( mem(V7v,ty_2Eextreal_2Eextreal)
=> ! [V8v1] :
( mem(V8v1,ty_2Eextreal_2Eextreal)
=> p(ap(ap(V0P,V7v),V8v1)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__sub__def,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2a] :
( mem(V2a,ty_2Eextreal_2Eextreal)
=> ! [V3v2] :
( mem(V3v2,ty_2Erealax_2Ereal)
=> ! [V4v5] :
( mem(V4v5,ty_2Erealax_2Ereal)
=> ! [V5v3] :
( mem(V5v3,ty_2Erealax_2Ereal)
=> ( ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2ENormal,V0x)),ap(c_2Eextreal_2ENormal,V1y)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal_2Ereal__sub,V0x),V1y))
& ap(ap(c_2Eextreal_2Eextreal__sub,c_2Eextreal_2EPosInf),V2a) = c_2Eextreal_2EPosInf
& ap(ap(c_2Eextreal_2Eextreal__sub,c_2Eextreal_2ENegInf),c_2Eextreal_2EPosInf) = c_2Eextreal_2ENegInf
& ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2ENormal,V3v2)),c_2Eextreal_2EPosInf) = c_2Eextreal_2ENegInf
& ap(ap(c_2Eextreal_2Eextreal__sub,c_2Eextreal_2ENegInf),c_2Eextreal_2ENegInf) = c_2Eextreal_2EPosInf
& ap(ap(c_2Eextreal_2Eextreal__sub,c_2Eextreal_2ENegInf),ap(c_2Eextreal_2ENormal,V4v5)) = c_2Eextreal_2ENegInf
& ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2ENormal,V5v3)),c_2Eextreal_2ENegInf) = c_2Eextreal_2EPosInf ) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__le__ind,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool)))
=> ( ( ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,ap(c_2Eextreal_2ENormal,V1x)),ap(c_2Eextreal_2ENormal,V2y))) ) )
& ! [V3a] :
( mem(V3a,ty_2Eextreal_2Eextreal)
=> p(ap(ap(V0P,c_2Eextreal_2ENegInf),V3a)) )
& p(ap(ap(V0P,c_2Eextreal_2EPosInf),c_2Eextreal_2EPosInf))
& ! [V4v2] :
( mem(V4v2,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,ap(c_2Eextreal_2ENormal,V4v2)),c_2Eextreal_2EPosInf)) )
& p(ap(ap(V0P,c_2Eextreal_2EPosInf),c_2Eextreal_2ENegInf))
& ! [V5v3] :
( mem(V5v3,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,ap(c_2Eextreal_2ENormal,V5v3)),c_2Eextreal_2ENegInf)) )
& ! [V6v5] :
( mem(V6v5,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,c_2Eextreal_2EPosInf),ap(c_2Eextreal_2ENormal,V6v5))) ) )
=> ! [V7v] :
( mem(V7v,ty_2Eextreal_2Eextreal)
=> ! [V8v1] :
( mem(V8v1,ty_2Eextreal_2Eextreal)
=> p(ap(ap(V0P,V7v),V8v1)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__le__def,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2a] :
( mem(V2a,ty_2Eextreal_2Eextreal)
=> ! [V3v2] :
( mem(V3v2,ty_2Erealax_2Ereal)
=> ! [V4v3] :
( mem(V4v3,ty_2Erealax_2Ereal)
=> ! [V5v5] :
( mem(V5v5,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2ENormal,V0x)),ap(c_2Eextreal_2ENormal,V1y)))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y)) )
& ( p(ap(ap(c_2Eextreal_2Eextreal__le,c_2Eextreal_2ENegInf),V2a))
<=> $true )
& ( p(ap(ap(c_2Eextreal_2Eextreal__le,c_2Eextreal_2EPosInf),c_2Eextreal_2EPosInf))
<=> $true )
& ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2ENormal,V3v2)),c_2Eextreal_2EPosInf))
<=> $true )
& ( p(ap(ap(c_2Eextreal_2Eextreal__le,c_2Eextreal_2EPosInf),c_2Eextreal_2ENegInf))
<=> $false )
& ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2ENormal,V4v3)),c_2Eextreal_2ENegInf))
<=> $false )
& ( p(ap(ap(c_2Eextreal_2Eextreal__le,c_2Eextreal_2EPosInf),ap(c_2Eextreal_2ENormal,V5v5)))
<=> $false ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__lt__def,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y))
<=> ~ p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__mul__ind,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool)))
=> ( ( p(ap(ap(V0P,c_2Eextreal_2ENegInf),c_2Eextreal_2ENegInf))
& p(ap(ap(V0P,c_2Eextreal_2ENegInf),c_2Eextreal_2EPosInf))
& p(ap(ap(V0P,c_2Eextreal_2EPosInf),c_2Eextreal_2ENegInf))
& p(ap(ap(V0P,c_2Eextreal_2EPosInf),c_2Eextreal_2EPosInf))
& ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,ap(c_2Eextreal_2ENormal,V1x)),c_2Eextreal_2ENegInf)) )
& ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,c_2Eextreal_2ENegInf),ap(c_2Eextreal_2ENormal,V2y))) )
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,ap(c_2Eextreal_2ENormal,V3x)),c_2Eextreal_2EPosInf)) )
& ! [V4y] :
( mem(V4y,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,c_2Eextreal_2EPosInf),ap(c_2Eextreal_2ENormal,V4y))) )
& ! [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
=> ! [V6y] :
( mem(V6y,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,ap(c_2Eextreal_2ENormal,V5x)),ap(c_2Eextreal_2ENormal,V6y))) ) ) )
=> ! [V7v] :
( mem(V7v,ty_2Eextreal_2Eextreal)
=> ! [V8v1] :
( mem(V8v1,ty_2Eextreal_2Eextreal)
=> p(ap(ap(V0P,V7v),V8v1)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__mul__def,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ap(ap(c_2Eextreal_2Eextreal__mul,c_2Eextreal_2ENegInf),c_2Eextreal_2ENegInf) = c_2Eextreal_2EPosInf
& ap(ap(c_2Eextreal_2Eextreal__mul,c_2Eextreal_2ENegInf),c_2Eextreal_2EPosInf) = c_2Eextreal_2ENegInf
& ap(ap(c_2Eextreal_2Eextreal__mul,c_2Eextreal_2EPosInf),c_2Eextreal_2ENegInf) = c_2Eextreal_2ENegInf
& ap(ap(c_2Eextreal_2Eextreal__mul,c_2Eextreal_2EPosInf),c_2Eextreal_2EPosInf) = c_2Eextreal_2EPosInf
& ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V0x)),c_2Eextreal_2ENegInf) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x)),c_2Eextreal_2ENegInf),c_2Eextreal_2EPosInf))
& ap(ap(c_2Eextreal_2Eextreal__mul,c_2Eextreal_2ENegInf),ap(c_2Eextreal_2ENormal,V1y)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),V1y),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1y)),c_2Eextreal_2ENegInf),c_2Eextreal_2EPosInf))
& ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V0x)),c_2Eextreal_2EPosInf) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x)),c_2Eextreal_2EPosInf),c_2Eextreal_2ENegInf))
& ap(ap(c_2Eextreal_2Eextreal__mul,c_2Eextreal_2EPosInf),ap(c_2Eextreal_2ENormal,V1y)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),V1y),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1y)),c_2Eextreal_2EPosInf),c_2Eextreal_2ENegInf))
& ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V0x)),ap(c_2Eextreal_2ENormal,V1y)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)) ) ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__inv__def,axiom,
( ap(c_2Eextreal_2Eextreal__inv,c_2Eextreal_2ENegInf) = ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))
& ap(c_2Eextreal_2Eextreal__inv,c_2Eextreal_2EPosInf) = ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))
& ! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Eextreal_2Eextreal__inv,ap(c_2Eextreal_2ENormal,V0x)) = ap(c_2Eextreal_2ENormal,ap(c_2Erealax_2Einv,V0x)) ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__ainv__def,axiom,
( ap(c_2Eextreal_2Eextreal__ainv,c_2Eextreal_2ENegInf) = c_2Eextreal_2EPosInf
& ap(c_2Eextreal_2Eextreal__ainv,c_2Eextreal_2EPosInf) = c_2Eextreal_2ENegInf
& ! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2ENormal,V0x)) = ap(c_2Eextreal_2ENormal,ap(c_2Erealax_2Ereal__neg,V0x)) ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__div__def,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__div,V0x),V1y) = ap(ap(c_2Eextreal_2Eextreal__mul,V0x),ap(c_2Eextreal_2Eextreal__inv,V1y)) ) ) ).
fof(lameq_f3347,axiom,
! [V0R] : ap(f3347,V0R) = ap(c_2Erelation_2EWF(ty_2Eextreal_2Eextreal),V0R) ).
fof(lameq_f3348,axiom,
! [V3x] : ap(f3348,V3x) = ap(c_2Ecombin_2EI(ty_2Eextreal_2Eextreal),ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Eabs,V3x))) ).
fof(lameq_f3349,axiom,
! [V2a] : ap(f3349,V2a) = ap(ap(ap(ap(c_2Eextreal_2Eextreal__CASE(ty_2Eextreal_2Eextreal),V2a),ap(c_2Ecombin_2EI(ty_2Eextreal_2Eextreal),c_2Eextreal_2EPosInf)),ap(c_2Ecombin_2EI(ty_2Eextreal_2Eextreal),c_2Eextreal_2EPosInf)),f3348) ).
fof(ax_thm_2Eextreal_2Eextreal__abs__primitive__def,axiom,
c_2Eextreal_2Eextreal__abs = ap(ap(c_2Erelation_2EWFREC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),ap(c_2Emin_2E_40(arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool))),f3347)),k(arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3349)) ).
fof(conj_thm_2Eextreal_2Eextreal__abs__ind,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,bool))
=> ( ( ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> p(ap(V0P,ap(c_2Eextreal_2ENormal,V1x))) )
& p(ap(V0P,c_2Eextreal_2ENegInf))
& p(ap(V0P,c_2Eextreal_2EPosInf)) )
=> ! [V2v] :
( mem(V2v,ty_2Eextreal_2Eextreal)
=> p(ap(V0P,V2v)) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__abs__def,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ( ap(c_2Eextreal_2Eextreal__abs,ap(c_2Eextreal_2ENormal,V0x)) = ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Eabs,V0x))
& ap(c_2Eextreal_2Eextreal__abs,c_2Eextreal_2ENegInf) = c_2Eextreal_2EPosInf
& ap(c_2Eextreal_2Eextreal__abs,c_2Eextreal_2EPosInf) = c_2Eextreal_2EPosInf ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__logr__def,axiom,
( ! [V0b] :
( mem(V0b,ty_2Erealax_2Ereal)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Eextreal_2Eextreal__logr,V0b),ap(c_2Eextreal_2ENormal,V1x)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Eutil__prob_2Elogr,V0b),V1x)) ) )
& ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ap(ap(c_2Eextreal_2Eextreal__logr,V2b),c_2Eextreal_2EPosInf) = c_2Eextreal_2EPosInf )
& ! [V3b] :
( mem(V3b,ty_2Erealax_2Ereal)
=> ap(ap(c_2Eextreal_2Eextreal__logr,V3b),c_2Eextreal_2ENegInf) = c_2Eextreal_2ENegInf ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__lg__def,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(c_2Eextreal_2Eextreal__lg,V0x) = ap(ap(c_2Eextreal_2Eextreal__logr,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V0x) ) ).
fof(ax_thm_2Eextreal_2Eextreal__exp__def,axiom,
( ! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Eextreal_2Eextreal__exp,ap(c_2Eextreal_2ENormal,V0x)) = ap(c_2Eextreal_2ENormal,ap(c_2Etransc_2Eexp,V0x)) )
& ap(c_2Eextreal_2Eextreal__exp,c_2Eextreal_2EPosInf) = c_2Eextreal_2EPosInf
& ap(c_2Eextreal_2Eextreal__exp,c_2Eextreal_2ENegInf) = ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ).
fof(ax_thm_2Eextreal_2Eextreal__pow__def,axiom,
( ! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2ENormal,V0a)),V1n) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal_2Epow,V0a),V1n)) ) )
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Eextreal_2Eextreal__pow,c_2Eextreal_2EPosInf),V2n) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V2n),c_2Enum_2E0)),ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),c_2Eextreal_2EPosInf) )
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(c_2Eextreal_2Eextreal__pow,c_2Eextreal_2ENegInf),V3n) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V3n),c_2Enum_2E0)),ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(c_2Earithmetic_2EEVEN,V3n)),c_2Eextreal_2EPosInf),c_2Eextreal_2ENegInf)) ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__sqrt__def,axiom,
( ! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Eextreal_2Eextreal__sqrt,ap(c_2Eextreal_2ENormal,V0x)) = ap(c_2Eextreal_2ENormal,ap(c_2Etransc_2Esqrt,V0x)) )
& ap(c_2Eextreal_2Eextreal__sqrt,c_2Eextreal_2EPosInf) = c_2Eextreal_2EPosInf ) ).
fof(conj_thm_2Eextreal_2EREAL__LT__LMUL__0__NEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)))
& p(ap(ap(c_2Erealax_2Ereal__lt,V0x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,V1y),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ).
fof(conj_thm_2Eextreal_2EREAL__LT__RMUL__0__NEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)))
& p(ap(ap(c_2Erealax_2Ereal__lt,V1y),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,V0x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ).
fof(conj_thm_2Eextreal_2EREAL__LT__LMUL__NEG__0,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x)) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,V1y),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ).
fof(conj_thm_2Eextreal_2EREAL__LT__RMUL__NEG__0,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1y)) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,V0x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ).
fof(conj_thm_2Eextreal_2EREAL__LT__LMUL__NEG__0__NEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
& p(ap(ap(c_2Erealax_2Ereal__lt,V0x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1y)) ) ) ) ).
fof(conj_thm_2Eextreal_2EREAL__LT__RMUL__NEG__0__NEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
& p(ap(ap(c_2Erealax_2Ereal__lt,V1y),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x)) ) ) ) ).
fof(conj_thm_2Eextreal_2EREAL__LT__RDIV__EQ__NEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V2z),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Ereal_2E_2F,V1y),V2z)),V0x))
<=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__mul,V0x),V2z)),V1y)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EREAL__LE__RDIV__EQ__NEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V2z),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2E_2F,V1y),V2z)),V0x))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,V0x),V2z)),V1y)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EPOW__POS__EVEN,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V1x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2Epow,V1x),V0n)))
<=> p(ap(c_2Earithmetic_2EEVEN,V0n)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EPOW__NEG__ODD,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V1x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Ereal_2Epow,V1x),V0n)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
<=> p(ap(c_2Earithmetic_2EODD,V0n)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2ELOGR__MONO__LE,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1y))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V2b)) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eutil__prob_2Elogr,V2b),V0x)),ap(ap(c_2Eutil__prob_2Elogr,V2b),V1y)))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2ELOGR__MONO__LE__IMP,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V2b)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eutil__prob_2Elogr,V2b),V0x)),ap(ap(c_2Eutil__prob_2Elogr,V2b),V1y))) ) ) ) ) ).
fof(ax_thm_2Eextreal_2Emono__increasing__def,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eextreal_2Emono__increasing,V0f))
<=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V1m)),ap(V0f,V2n))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Emono__increasing__suc,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eextreal_2Emono__increasing,V0f))
<=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V1n)),ap(V0f,ap(c_2Enum_2ESUC,V1n)))) ) ) ) ).
fof(ax_thm_2Eextreal_2Emono__decreasing__def,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eextreal_2Emono__decreasing,V0f))
<=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V2n)),ap(V0f,V1m))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Emono__decreasing__suc,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eextreal_2Emono__decreasing,V0f))
<=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,ap(c_2Enum_2ESUC,V1n))),ap(V0f,V1n))) ) ) ) ).
fof(conj_thm_2Eextreal_2Emono__increasing__converges__to__sup,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1r] :
( mem(V1r,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Eextreal_2Emono__increasing,V0f))
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V1r)) )
=> V1r = ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Erealax_2Ereal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__cases,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( V0x = c_2Eextreal_2ENegInf
| V0x = c_2Eextreal_2EPosInf
| ? [V1r] :
( mem(V1r,ty_2Erealax_2Ereal)
& V0x = ap(c_2Eextreal_2ENormal,V1r) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__eq__zero,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ( ap(c_2Eextreal_2ENormal,V0x) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)
<=> V0x = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__not__infty,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ( ap(c_2Eextreal_2ENormal,V0x) != c_2Eextreal_2ENegInf
& ap(c_2Eextreal_2ENormal,V0x) != c_2Eextreal_2EPosInf ) ) ).
fof(conj_thm_2Eextreal_2Enum__not__infty,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ( ap(c_2Eextreal_2Eextreal__of__num,V0n) != c_2Eextreal_2ENegInf
& ap(c_2Eextreal_2Eextreal__of__num,V0n) != c_2Eextreal_2EPosInf ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__11,axiom,
! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ! [V1a_27] :
( mem(V1a_27,ty_2Erealax_2Ereal)
=> ( ap(c_2Eextreal_2ENormal,V0a) = ap(c_2Eextreal_2ENormal,V1a_27)
<=> V0a = V1a_27 ) ) ) ).
fof(conj_thm_2Eextreal_2Emul__rzero,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__mul,V0x),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Eextreal_2Emul__lzero,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Eextreal_2Emul__rone,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__mul,V0x),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) = V0x ) ).
fof(conj_thm_2Eextreal_2Emul__lone,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x) = V0x ) ).
fof(conj_thm_2Eextreal_2Eentire,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)
<=> ( V0x = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)
| V1y = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__lt__eq,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2ENormal,V0x)),ap(c_2Eextreal_2ENormal,V1y)))
<=> p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V1y)) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__refl,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V0x)) ) ).
fof(conj_thm_2Eextreal_2Elt__refl,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ~ p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V0x)) ) ).
fof(conj_thm_2Eextreal_2Ele__infty,axiom,
( ! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,c_2Eextreal_2ENegInf),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),c_2Eextreal_2EPosInf)) ) )
& ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),c_2Eextreal_2ENegInf))
<=> V1x = c_2Eextreal_2ENegInf ) )
& ! [V2x] :
( mem(V2x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,c_2Eextreal_2EPosInf),V2x))
<=> V2x = c_2Eextreal_2EPosInf ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__infty,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,c_2Eextreal_2ENegInf),ap(c_2Eextreal_2ENormal,V1y)))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2ENormal,V1y)),c_2Eextreal_2EPosInf))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,c_2Eextreal_2ENegInf),c_2Eextreal_2EPosInf))
& ~ p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),c_2Eextreal_2ENegInf))
& ~ p(ap(ap(c_2Eextreal_2Eextreal__lt,c_2Eextreal_2EPosInf),V0x))
& ( V0x != c_2Eextreal_2EPosInf
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),c_2Eextreal_2EPosInf)) )
& ( V0x != c_2Eextreal_2ENegInf
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,c_2Eextreal_2ENegInf),V0x)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__imp__le,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y)) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__imp__ne,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y))
=> V0x != V1y ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__trans,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),V2z)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__trans,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),V2z)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elet__trans,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),V2z)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__antisym,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),V0x)) )
<=> V0x = V1y ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__antisym,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ~ ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Eextreal_2Elte__trans,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),V2z)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__total,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y))
| p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__total,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( V0x = V1y
| p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y))
| p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__01,axiom,
p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ).
fof(conj_thm_2Eextreal_2Elt__01,axiom,
p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ).
fof(conj_thm_2Eextreal_2Ene__01,axiom,
ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) != ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ).
fof(conj_thm_2Eextreal_2Ele__02,axiom,
p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) ).
fof(conj_thm_2Eextreal_2Elt__02,axiom,
p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) ).
fof(conj_thm_2Eextreal_2Ene__02,axiom,
ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) != ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) ).
fof(conj_thm_2Eextreal_2Ele__num,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(c_2Eextreal_2Eextreal__of__num,V0n))) ) ).
fof(conj_thm_2Eextreal_2Elt__le,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y))
<=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y))
& V0x != V1y ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__lt,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y))
<=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y))
| V0x = V1y ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__neg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__ainv,V0x)),ap(c_2Eextreal_2Eextreal__ainv,V1y)))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__neg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__ainv,V0x)),ap(c_2Eextreal_2Eextreal__ainv,V1y)))
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__add,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__add,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Elet__add,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Elte__add,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__add2,axiom,
! [V0w] :
( mem(V0w,ty_2Eextreal_2Eextreal)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ! [V3z] :
( mem(V3z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0w),V1x))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V2y),V3z)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__add,V0w),V2y)),ap(ap(c_2Eextreal_2Eextreal__add,V1x),V3z))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__add2,axiom,
! [V0w] :
( mem(V0w,ty_2Eextreal_2Eextreal)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ! [V3z] :
( mem(V3z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0w),V1x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V2y),V3z)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__add,V0w),V2y)),ap(ap(c_2Eextreal_2Eextreal__add,V1x),V3z))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elet__add2,axiom,
! [V0w] :
( mem(V0w,ty_2Eextreal_2Eextreal)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ! [V3z] :
( mem(V3z,ty_2Eextreal_2Eextreal)
=> ( ( V0w != c_2Eextreal_2ENegInf
& V0w != c_2Eextreal_2EPosInf
& p(ap(ap(c_2Eextreal_2Eextreal__le,V0w),V1x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V2y),V3z)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__add,V0w),V2y)),ap(ap(c_2Eextreal_2Eextreal__add,V1x),V3z))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elet__add2__alt,axiom,
! [V0w] :
( mem(V0w,ty_2Eextreal_2Eextreal)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ! [V3z] :
( mem(V3z,ty_2Eextreal_2Eextreal)
=> ( ( V1x != c_2Eextreal_2ENegInf
& V1x != c_2Eextreal_2EPosInf
& p(ap(ap(c_2Eextreal_2Eextreal__le,V0w),V1x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V2y),V3z)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__add,V0w),V2y)),ap(ap(c_2Eextreal_2Eextreal__add,V1x),V3z))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__addr,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y)))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__addr__imp,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__ladd,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y)),ap(ap(c_2Eextreal_2Eextreal__add,V0x),V2z)))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),V2z)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__radd,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__add,V1y),V0x)),ap(ap(c_2Eextreal_2Eextreal__add,V2z),V0x)))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),V2z)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__radd__imp,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),V2z))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__add,V1y),V0x)),ap(ap(c_2Eextreal_2Eextreal__add,V2z),V0x))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__ladd__imp,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),V2z))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y)),ap(ap(c_2Eextreal_2Eextreal__add,V0x),V2z))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__ladd,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y)),ap(ap(c_2Eextreal_2Eextreal__add,V0x),V2z)))
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),V2z)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__radd,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__add,V1y),V0x)),ap(ap(c_2Eextreal_2Eextreal__add,V2z),V0x)))
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),V2z)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__addl,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y)))
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__lneg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__ainv,V0x)),V1y))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__mul,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Elet__mul,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Elte__mul,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__mul__neg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Emul__le,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__mul,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__mul__neg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Emul__lt,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) ) ) ) ).
fof(conj_thm_2Eextreal_2Emul__let,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) ) ) ) ).
fof(conj_thm_2Eextreal_2Emul__lte,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__rmul__imp,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V2z))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V2z)),ap(ap(c_2Eextreal_2Eextreal__mul,V1y),V2z))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__lmul__imp,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V2z))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__mul,V2z),V0x)),ap(ap(c_2Eextreal_2Eextreal__mul,V2z),V1y))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__rmul,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V2z))
& V2z != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V2z)),ap(ap(c_2Eextreal_2Eextreal__mul,V1y),V2z)))
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__lmul,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& V0x != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)),ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V2z)))
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),V2z)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__mul2,axiom,
! [V0x1] :
( mem(V0x1,ty_2Eextreal_2Eextreal)
=> ! [V1x2] :
( mem(V1x2,ty_2Eextreal_2Eextreal)
=> ! [V2y1] :
( mem(V2y1,ty_2Eextreal_2Eextreal)
=> ! [V3y2] :
( mem(V3y2,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x1))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V2y1))
& V0x1 != c_2Eextreal_2EPosInf
& V2y1 != c_2Eextreal_2EPosInf
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x1),V1x2))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V2y1),V3y2)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__mul,V0x1),V2y1)),ap(ap(c_2Eextreal_2Eextreal__mul,V1x2),V3y2))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eabs__pos,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(c_2Eextreal_2Eextreal__abs,V0x))) ) ).
fof(conj_thm_2Eextreal_2Eabs__refl,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( ap(c_2Eextreal_2Eextreal__abs,V0x) = V0x
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x)) ) ) ).
fof(conj_thm_2Eextreal_2Eabs__bounds,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1k] :
( mem(V1k,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__abs,V0x)),V1k))
<=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__ainv,V1k)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1k)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eabs__bounds__lt,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1k] :
( mem(V1k,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__abs,V0x)),V1k))
<=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__ainv,V1k)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1k)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__comm,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y) = ap(ap(c_2Eextreal_2Eextreal__add,V1y),V0x) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__assoc,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__add,V0x),ap(ap(c_2Eextreal_2Eextreal__add,V1y),V2z)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y)),V2z) ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__not__infty,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( ( V0x != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2ENegInf )
=> ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y) != c_2Eextreal_2ENegInf )
& ( ( V0x != c_2Eextreal_2EPosInf
& V1y != c_2Eextreal_2EPosInf )
=> ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y) != c_2Eextreal_2EPosInf ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__rzero,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__add,V0x),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)) = V0x ) ).
fof(conj_thm_2Eextreal_2Eadd__lzero,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x) = V0x ) ).
fof(conj_thm_2Eextreal_2Eadd__infty,axiom,
( ! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( ap(ap(c_2Eextreal_2Eextreal__add,V0x),c_2Eextreal_2EPosInf) = c_2Eextreal_2EPosInf
& ap(ap(c_2Eextreal_2Eextreal__add,c_2Eextreal_2EPosInf),V0x) = c_2Eextreal_2EPosInf ) )
& ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( V1x != c_2Eextreal_2EPosInf
=> ( ap(ap(c_2Eextreal_2Eextreal__add,V1x),c_2Eextreal_2ENegInf) = c_2Eextreal_2ENegInf
& ap(ap(c_2Eextreal_2Eextreal__add,c_2Eextreal_2ENegInf),V1x) = c_2Eextreal_2ENegInf ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__EQ__LADD,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
=> ( ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y) = ap(ap(c_2Eextreal_2Eextreal__add,V0x),V2z)
<=> V1y = V2z ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__rzero,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__sub,V0x),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)) = V0x ) ).
fof(conj_thm_2Eextreal_2Esub__lzero,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x) = ap(c_2Eextreal_2Eextreal__ainv,V0x) ) ).
fof(conj_thm_2Eextreal_2Esub__le__imp,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf
& p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(ap(c_2Eextreal_2Eextreal__add,V2z),V0x))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x)),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__le__imp2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf
& p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(ap(c_2Eextreal_2Eextreal__add,V2z),V0x))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x)),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__sub__imp,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__add,V1y),V0x)),V2z))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(ap(c_2Eextreal_2Eextreal__sub,V2z),V0x))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__sub__imp,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__add,V1y),V0x)),V2z))
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),ap(ap(c_2Eextreal_2Eextreal__sub,V2z),V0x))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__lt__imp,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),ap(ap(c_2Eextreal_2Eextreal__add,V2z),V0x))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x)),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__lt__imp2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V2z != c_2Eextreal_2ENegInf
& V2z != c_2Eextreal_2EPosInf
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),ap(ap(c_2Eextreal_2Eextreal__add,V2z),V0x))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x)),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__zero__lt,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y))
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x))) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__zero__lt2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y)) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__lt__zero,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y))
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__lt__zero2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y)) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__zero__le,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x))) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__le__zero,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__sub__eq,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(ap(c_2Eextreal_2Eextreal__sub,V2z),V0x)))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__add,V1y),V0x)),V2z)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__sub__eq2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V2z != c_2Eextreal_2ENegInf
& V2z != c_2Eextreal_2EPosInf
& V0x != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2ENegInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(ap(c_2Eextreal_2Eextreal__sub,V2z),V0x)))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__add,V1y),V0x)),V2z)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__le__eq,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x)),V2z))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(ap(c_2Eextreal_2Eextreal__add,V2z),V0x))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__le__eq2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf
& V0x != c_2Eextreal_2ENegInf
& V2z != c_2Eextreal_2ENegInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x)),V2z))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(ap(c_2Eextreal_2Eextreal__add,V2z),V0x))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__le__switch,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf
& V2z != c_2Eextreal_2ENegInf
& V2z != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x)),V2z))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V2z)),V0x)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__le__switch2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf
& V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x)),V2z))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V2z)),V0x)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__sub,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V2z != c_2Eextreal_2ENegInf
& V2z != c_2Eextreal_2EPosInf )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__add,V1y),V0x)),V2z))
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),ap(ap(c_2Eextreal_2Eextreal__sub,V2z),V0x))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__add2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
=> ap(ap(c_2Eextreal_2Eextreal__add,V0x),ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x)) = V1y ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__sub,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf )
=> ap(ap(c_2Eextreal_2Eextreal__sub,ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y)),V1y) = V0x ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__sub2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf )
=> ap(ap(c_2Eextreal_2Eextreal__sub,ap(ap(c_2Eextreal_2Eextreal__add,V1y),V0x)),V1y) = V0x ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__add,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf )
=> ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y)),V1y) = V0x ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__sub__add,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y) = ap(ap(c_2Eextreal_2Eextreal__add,V0x),ap(c_2Eextreal_2Eextreal__ainv,V1y)) ) ) ).
fof(conj_thm_2Eextreal_2Esub__0,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)
=> V0x = V1y ) ) ) ).
fof(conj_thm_2Eextreal_2Eneg__neg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2Eextreal__ainv,V0x)) = V0x ) ).
fof(conj_thm_2Eextreal_2Eneg__0,axiom,
ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ).
fof(conj_thm_2Eextreal_2Eneg__eq0,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( ap(c_2Eextreal_2Eextreal__ainv,V0x) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)
<=> V0x = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ).
fof(conj_thm_2Eextreal_2Eeq__neg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ap(c_2Eextreal_2Eextreal__ainv,V0x) = ap(c_2Eextreal_2Eextreal__ainv,V1y)
<=> V0x = V1y ) ) ) ).
fof(conj_thm_2Eextreal_2Eneg__minus1,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(c_2Eextreal_2Eextreal__ainv,V0x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V0x) ) ).
fof(conj_thm_2Eextreal_2Esub__rneg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__sub,V0x),ap(c_2Eextreal_2Eextreal__ainv,V1y)) = ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y) ) ) ).
fof(conj_thm_2Eextreal_2Esub__lneg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( ( V0x != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2ENegInf )
| ( V0x != c_2Eextreal_2EPosInf
& V1y != c_2Eextreal_2EPosInf ) )
=> ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2Eextreal__ainv,V0x)),V1y) = ap(c_2Eextreal_2Eextreal__ainv,ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y)) ) ) ) ).
fof(conj_thm_2Eextreal_2Eneg__sub,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
| ( V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf ) )
=> ap(c_2Eextreal_2Eextreal__ainv,ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y)) = ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V0x) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__not__infty,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( ( V0x != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf )
=> ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y) != c_2Eextreal_2ENegInf )
& ( ( V0x != c_2Eextreal_2EPosInf
& V1y != c_2Eextreal_2ENegInf )
=> ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y) != c_2Eextreal_2EPosInf ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__lsub__imp,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),V2z))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V2z)),ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eeq__sub__ladd__normal,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( V0x = ap(ap(c_2Eextreal_2Eextreal__sub,V1y),ap(c_2Eextreal_2ENormal,V2z))
<=> ap(ap(c_2Eextreal_2Eextreal__add,V0x),ap(c_2Eextreal_2ENormal,V2z)) = V1y ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eeq__sub__radd,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf )
=> ( ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y) = V2z
<=> V0x = ap(ap(c_2Eextreal_2Eextreal__add,V2z),V1y) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eeq__sub__ladd,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V2z != c_2Eextreal_2ENegInf
& V2z != c_2Eextreal_2EPosInf )
=> ( V0x = ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V2z)
<=> ap(ap(c_2Eextreal_2Eextreal__add,V0x),V2z) = V1y ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eeq__sub__switch,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( V0x = ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2ENormal,V2z)),V1y)
<=> V1y = ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2ENormal,V2z)),V0x) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eeq__add__sub__switch,axiom,
! [V0a] :
( mem(V0a,ty_2Eextreal_2Eextreal)
=> ! [V1b] :
( mem(V1b,ty_2Eextreal_2Eextreal)
=> ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V3d] :
( mem(V3d,ty_2Eextreal_2Eextreal)
=> ( ( V1b != c_2Eextreal_2ENegInf
& V1b != c_2Eextreal_2EPosInf
& V2c != c_2Eextreal_2ENegInf
& V2c != c_2Eextreal_2EPosInf )
=> ( ap(ap(c_2Eextreal_2Eextreal__add,V0a),V1b) = ap(ap(c_2Eextreal_2Eextreal__add,V2c),V3d)
<=> ap(ap(c_2Eextreal_2Eextreal__sub,V0a),V2c) = ap(ap(c_2Eextreal_2Eextreal__sub,V3d),V1b) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__refl,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
=> ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V0x) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ).
fof(conj_thm_2Eextreal_2Emul__comm,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y) = ap(ap(c_2Eextreal_2Eextreal__mul,V1y),V0x) ) ) ).
fof(conj_thm_2Eextreal_2Emul__assoc,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__mul,V0x),ap(ap(c_2Eextreal_2Eextreal__mul,V1y),V2z)) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)),V2z) ) ) ) ).
fof(conj_thm_2Eextreal_2Emul__not__infty,axiom,
( ! [V0c] :
( mem(V0c,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0c))
& V1y != c_2Eextreal_2ENegInf )
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V0c)),V1y) != c_2Eextreal_2ENegInf ) ) )
& ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ! [V3y] :
( mem(V3y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2c))
& V3y != c_2Eextreal_2EPosInf )
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V2c)),V3y) != c_2Eextreal_2EPosInf ) ) )
& ! [V4c] :
( mem(V4c,ty_2Erealax_2Ereal)
=> ! [V5y] :
( mem(V5y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V4c),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
& V5y != c_2Eextreal_2ENegInf )
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V4c)),V5y) != c_2Eextreal_2EPosInf ) ) )
& ! [V6c] :
( mem(V6c,ty_2Erealax_2Ereal)
=> ! [V7y] :
( mem(V7y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V6c),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
& V7y != c_2Eextreal_2EPosInf )
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V6c)),V7y) != c_2Eextreal_2ENegInf ) ) ) ) ).
fof(conj_thm_2Eextreal_2Emul__not__infty2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf
& V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf )
=> ( ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y) != c_2Eextreal_2ENegInf
& ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y) != c_2Eextreal_2EPosInf ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__ldistrib__pos,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V2z)) )
=> ap(ap(c_2Eextreal_2Eextreal__mul,V0x),ap(ap(c_2Eextreal_2Eextreal__add,V1y),V2z)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)),ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__ldistrib__neg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V2z),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) )
=> ap(ap(c_2Eextreal_2Eextreal__mul,V0x),ap(ap(c_2Eextreal_2Eextreal__add,V1y),V2z)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)),ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__ldistrib__normal,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( ( V1y != c_2Eextreal_2EPosInf
& V2z != c_2Eextreal_2EPosInf )
| ( V1y != c_2Eextreal_2ENegInf
& V2z != c_2Eextreal_2ENegInf ) )
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V0x)),ap(ap(c_2Eextreal_2Eextreal__add,V1y),V2z)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V0x)),V1y)),ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V0x)),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__ldistrib__normal2,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x))
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V0x)),ap(ap(c_2Eextreal_2Eextreal__add,V1y),V2z)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V0x)),V1y)),ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V0x)),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__rdistrib__normal,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( ( V1y != c_2Eextreal_2EPosInf
& V2z != c_2Eextreal_2EPosInf )
| ( V1y != c_2Eextreal_2ENegInf
& V2z != c_2Eextreal_2ENegInf ) )
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(c_2Eextreal_2Eextreal__add,V1y),V2z)),ap(c_2Eextreal_2ENormal,V0x)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__mul,V1y),ap(c_2Eextreal_2ENormal,V0x))),ap(ap(c_2Eextreal_2Eextreal__mul,V2z),ap(c_2Eextreal_2ENormal,V0x))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__rdistrib__normal2,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x))
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(c_2Eextreal_2Eextreal__add,V1y),V2z)),ap(c_2Eextreal_2ENormal,V0x)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__mul,V1y),ap(c_2Eextreal_2ENormal,V0x))),ap(ap(c_2Eextreal_2Eextreal__mul,V2z),ap(c_2Eextreal_2ENormal,V0x))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__ldistrib,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V2z)) )
| ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V2z),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) ) )
=> ap(ap(c_2Eextreal_2Eextreal__mul,V0x),ap(ap(c_2Eextreal_2Eextreal__add,V1y),V2z)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)),ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__rdistrib,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1y))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V2z)) )
| ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V2z),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) ) )
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(c_2Eextreal_2Eextreal__add,V1y),V2z)),V0x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__mul,V1y),V0x)),ap(ap(c_2Eextreal_2Eextreal__mul,V2z),V0x)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Emul__lneg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__ainv,V0x)),V1y) = ap(c_2Eextreal_2Eextreal__ainv,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)) ) ) ).
fof(conj_thm_2Eextreal_2Emul__rneg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__mul,V0x),ap(c_2Eextreal_2Eextreal__ainv,V1y)) = ap(c_2Eextreal_2Eextreal__ainv,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)) ) ) ).
fof(conj_thm_2Eextreal_2Eneg__mul2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__ainv,V0x)),ap(c_2Eextreal_2Eextreal__ainv,V1y)) = ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y) ) ) ).
fof(conj_thm_2Eextreal_2Eadd2__sub2,axiom,
! [V0a] :
( mem(V0a,ty_2Eextreal_2Eextreal)
=> ! [V1b] :
( mem(V1b,ty_2Eextreal_2Eextreal)
=> ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V3d] :
( mem(V3d,ty_2Eextreal_2Eextreal)
=> ( ( ( V1b != c_2Eextreal_2EPosInf
& V3d != c_2Eextreal_2EPosInf )
| ( V1b != c_2Eextreal_2ENegInf
& V3d != c_2Eextreal_2ENegInf ) )
=> ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__sub,V0a),V1b)),ap(ap(c_2Eextreal_2Eextreal__sub,V2c),V3d)) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(ap(c_2Eextreal_2Eextreal__add,V0a),V2c)),ap(ap(c_2Eextreal_2Eextreal__add,V1b),V3d)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__ldistrib,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf
& V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf
& V2z != c_2Eextreal_2ENegInf
& V2z != c_2Eextreal_2EPosInf )
=> ap(ap(c_2Eextreal_2Eextreal__mul,V0x),ap(ap(c_2Eextreal_2Eextreal__sub,V1y),V2z)) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)),ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esub__rdistrib,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf
& V1y != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2EPosInf
& V2z != c_2Eextreal_2ENegInf
& V2z != c_2Eextreal_2EPosInf )
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y)),V2z) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V2z)),ap(ap(c_2Eextreal_2Eextreal__mul,V1y),V2z)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eextreal__div__eq,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2ENormal,V0x)),ap(c_2Eextreal_2ENormal,V1y)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal_2E_2F,V0x),V1y)) ) ) ).
fof(conj_thm_2Eextreal_2Einv__one,axiom,
ap(c_2Eextreal_2Eextreal__inv,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) = ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ).
fof(conj_thm_2Eextreal_2Einv__1over,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(c_2Eextreal_2Eextreal__inv,V0x) = ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x) ) ).
fof(conj_thm_2Eextreal_2Ediv__one,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__div,V0x),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) = V0x ) ).
fof(conj_thm_2Eextreal_2Einv__pos,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& V0x != c_2Eextreal_2EPosInf )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x))) ) ) ).
fof(conj_thm_2Eextreal_2Erinv__uniq,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y) = ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))
=> V1y = ap(c_2Eextreal_2Eextreal__inv,V0x) ) ) ) ).
fof(conj_thm_2Eextreal_2Elinv__uniq,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y) = ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))
=> V0x = ap(c_2Eextreal_2Eextreal__inv,V1y) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__rdiv,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__mul,V1y),ap(c_2Eextreal_2ENormal,V0x))),V2z))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(ap(c_2Eextreal_2Eextreal__div,V2z),ap(c_2Eextreal_2ENormal,V0x)))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__ldiv,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(ap(c_2Eextreal_2Eextreal__mul,V2z),ap(c_2Eextreal_2ENormal,V0x))))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__div,V1y),ap(c_2Eextreal_2ENormal,V0x))),V2z)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__rdiv,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2z))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),ap(ap(c_2Eextreal_2Eextreal__div,V1y),ap(c_2Eextreal_2ENormal,V2z))))
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),ap(c_2Eextreal_2ENormal,V2z))),V1y)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__ldiv,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2z))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__div,V0x),ap(c_2Eextreal_2ENormal,V2z))),V1y))
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),ap(ap(c_2Eextreal_2Eextreal__mul,V1y),ap(c_2Eextreal_2ENormal,V2z)))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Elt__rdiv__neg,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V2z),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__div,V1y),ap(c_2Eextreal_2ENormal,V2z))),V0x))
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__mul,V0x),ap(c_2Eextreal_2ENormal,V2z))),V1y)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ediv__add,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V1y != c_2Eextreal_2ENegInf
& V2z != ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) )
=> ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__div,V0x),V2z)),ap(ap(c_2Eextreal_2Eextreal__div,V1y),V2z)) = ap(ap(c_2Eextreal_2Eextreal__div,ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y)),V2z) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__inv,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(c_2Eextreal_2Eextreal__inv,V0x))) ) ) ).
fof(conj_thm_2Eextreal_2Epow__0,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__pow,V0x),c_2Enum_2E0) = ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ).
fof(conj_thm_2Eextreal_2Epow__1,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__pow,V0x),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) = V0x ) ).
fof(conj_thm_2Eextreal_2Epow__2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__pow,V0x),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) = ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V0x) ) ).
fof(conj_thm_2Eextreal_2Epow__zero,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( ap(ap(c_2Eextreal_2Eextreal__pow,V1x),ap(c_2Enum_2ESUC,V0n)) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)
<=> V1x = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Eextreal_2Epow__zero__imp,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( ap(ap(c_2Eextreal_2Eextreal__pow,V1x),V0n) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)
=> V1x = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__pow2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__pow,V0x),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) ) ).
fof(conj_thm_2Eextreal_2Epow__pos__le,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__pow,V0x),V1n))) ) ) ) ).
fof(conj_thm_2Eextreal_2Epow__pos__lt,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__pow,V0x),V1n))) ) ) ) ).
fof(conj_thm_2Eextreal_2Epow__le,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1x))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),V2y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__pow,V1x),V0n)),ap(ap(c_2Eextreal_2Eextreal__pow,V2y),V0n))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Epow__lt,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1x),V2y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__pow,V1x),ap(c_2Enum_2ESUC,V0n))),ap(ap(c_2Eextreal_2Eextreal__pow,V2y),ap(c_2Enum_2ESUC,V0n)))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Epow__lt2,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( ( V0n != c_2Enum_2E0
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1x),V2y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__pow,V1x),V0n)),ap(ap(c_2Eextreal_2Eextreal__pow,V2y),V0n))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Epow__le__mono,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),V2m)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__pow,V0x),V1n)),ap(ap(c_2Eextreal_2Eextreal__pow,V0x),V2m))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Epow__pos__even,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V1x),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__pow,V1x),V0n)))
<=> p(ap(c_2Earithmetic_2EEVEN,V0n)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Epow__neg__odd,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V1x),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__pow,V1x),V0n)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)))
<=> p(ap(c_2Earithmetic_2EODD,V0n)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eadd__pow2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__pow,ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__pow,V0x),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Eextreal_2Eextreal__pow,V1y),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V0x)),V1y)) ) ) ).
fof(conj_thm_2Eextreal_2Epow__add,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ap(ap(c_2Eextreal_2Eextreal__pow,V0x),ap(ap(c_2Earithmetic_2E_2B,V1n),V2m)) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(c_2Eextreal_2Eextreal__pow,V0x),V1n)),ap(ap(c_2Eextreal_2Eextreal__pow,V0x),V2m)) ) ) ) ).
fof(conj_thm_2Eextreal_2Epow__mul,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__pow,ap(ap(c_2Eextreal_2Eextreal__mul,V1x),V2y)),V0n) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(c_2Eextreal_2Eextreal__pow,V1x),V0n)),ap(ap(c_2Eextreal_2Eextreal__pow,V2y),V0n)) ) ) ) ).
fof(conj_thm_2Eextreal_2Epow__minus1,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V0n)) = ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ).
fof(conj_thm_2Eextreal_2Epow__not__infty,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( ( V1x != c_2Eextreal_2ENegInf
& V1x != c_2Eextreal_2EPosInf )
=> ( ap(ap(c_2Eextreal_2Eextreal__pow,V1x),V0n) != c_2Eextreal_2ENegInf
& ap(ap(c_2Eextreal_2Eextreal__pow,V1x),V0n) != c_2Eextreal_2EPosInf ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esqrt__pos__le,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(c_2Eextreal_2Eextreal__sqrt,V0x))) ) ) ).
fof(conj_thm_2Eextreal_2Esqrt__pos__lt,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(c_2Eextreal_2Eextreal__sqrt,V0x))) ) ) ).
fof(conj_thm_2Eextreal_2Epow2__sqrt,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
=> ap(c_2Eextreal_2Eextreal__sqrt,ap(ap(c_2Eextreal_2Eextreal__pow,V0x),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) = V0x ) ) ).
fof(conj_thm_2Eextreal_2Esqrt__pow2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__sqrt,V0x)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) = V0x
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x)) ) ) ).
fof(conj_thm_2Eextreal_2Esqrt__mono__le,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__sqrt,V0x)),ap(c_2Eextreal_2Eextreal__sqrt,V1y))) ) ) ) ).
fof(conj_thm_2Eextreal_2Elogr__not__infty,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
=> ( ( V0x != c_2Eextreal_2ENegInf
& V0x != c_2Eextreal_2EPosInf )
=> ( ap(ap(c_2Eextreal_2Eextreal__logr,V1b),V0x) != c_2Eextreal_2ENegInf
& ap(ap(c_2Eextreal_2Eextreal__logr,V1b),V0x) != c_2Eextreal_2EPosInf ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ehalf__between,axiom,
( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ).
fof(conj_thm_2Eextreal_2Ethirds__between,axiom,
( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ).
fof(conj_thm_2Eextreal_2Ehalf__cancel,axiom,
ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) = ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ).
fof(conj_thm_2Eextreal_2Ethird__cancel,axiom,
ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))) = ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ).
fof(conj_thm_2Eextreal_2Efourth__cancel,axiom,
ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))) = ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ).
fof(conj_thm_2Eextreal_2Equotient__normal,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,V0n)),ap(c_2Eextreal_2Eextreal__of__num,V1m)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,V0n)),ap(c_2Ereal_2Ereal__of__num,V1m))) ) ) ).
fof(ax_thm_2Eextreal_2Eext__mono__increasing__def,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( p(ap(c_2Eextreal_2Eext__mono__increasing,V0f))
<=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V1m)),ap(V0f,V2n))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eext__mono__increasing__suc,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( p(ap(c_2Eextreal_2Eext__mono__increasing,V0f))
<=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V1n)),ap(V0f,ap(c_2Enum_2ESUC,V1n)))) ) ) ) ).
fof(ax_thm_2Eextreal_2Eext__mono__decreasing__def,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( p(ap(c_2Eextreal_2Eext__mono__decreasing,V0f))
<=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V2n)),ap(V0f,V1m))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Eext__mono__decreasing__suc,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( p(ap(c_2Eextreal_2Eext__mono__decreasing,V0f))
<=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,ap(c_2Enum_2ESUC,V1n))),ap(V0f,V1n))) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__ARCH,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( V1y != c_2Eextreal_2EPosInf
=> ? [V2n] :
( mem(V2n,ty_2Enum_2Enum)
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__of__num,V2n)),V0x))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2ESIMP__REAL__ARCH,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& p(ap(ap(c_2Ereal_2Ereal__lte,V0x),ap(c_2Ereal_2Ereal__of__num,V1n))) ) ) ).
fof(conj_thm_2Eextreal_2ESIMP__REAL__ARCH__NEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Erealax_2Ereal__neg,ap(c_2Ereal_2Ereal__of__num,V1n))),V0x)) ) ) ).
fof(conj_thm_2Eextreal_2ESIMP__EXTREAL__ARCH,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( V0x != c_2Eextreal_2EPosInf
=> ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),ap(c_2Eextreal_2Eextreal__of__num,V1n))) ) ) ) ).
fof(conj_thm_2Eextreal_2EREAL__ARCH__POW,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& p(ap(ap(c_2Erealax_2Ereal__lt,V0x),ap(ap(c_2Ereal_2Epow,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V1n))) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__ARCH__POW,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( V0x != c_2Eextreal_2EPosInf
=> ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V1n))) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__ARCH__POW__INV,axiom,
! [V0e] :
( mem(V0e,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0e))
=> ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal_2Epow,ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),V1n))),V0e)) ) ) ) ).
fof(conj_thm_2Eextreal_2EREAL__LE__MUL__EPSILON,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2z))
& p(ap(ap(c_2Erealax_2Ereal__lt,V2z),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,V2z),V0x)),V1y)) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y)) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__epsilon,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ! [V2e] :
( mem(V2e,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V2e))
& V2e != c_2Eextreal_2EPosInf )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),ap(ap(c_2Eextreal_2Eextreal__add,V1y),V2e))) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y)) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__mul__epsilon,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V2z))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V2z),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__mul,V2z),V0x)),V1y)) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y)) ) ) ) ).
fof(lameq_f3350,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2e] :
( mem(V2e,A_27a)
=> ! [V3acc] : ap(f3350(A_27a,V0f,V2e),V3acc) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,V2e)),V3acc) ) ) ).
fof(lameq_f3351,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2e] : ap(f3351(A_27a,V0f),V2e) = f3350(A_27a,V0f,V2e) ) ).
fof(ax_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V1s) = ap(ap(ap(c_2Epred__set_2EITSET(A_27a,ty_2Eextreal_2Eextreal),f3351(A_27a,V0f)),V1s),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),c_2Epred__set_2EEMPTY(A_27a)) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)
& ! [V1e] :
( mem(V1e,A_27a)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),V2s)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,V1e)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2s),V1e))) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NOT__NEG__INF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ap(V0f,V2x) != c_2Eextreal_2ENegInf ) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V1s) != c_2Eextreal_2ENegInf ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NOT__POS__INF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ap(V0f,V2x) != c_2Eextreal_2EPosInf ) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V1s) != c_2Eextreal_2EPosInf ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NOT__INFTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ap(V0f,V2x) != c_2Eextreal_2ENegInf ) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V1s) != c_2Eextreal_2ENegInf )
& ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
=> ap(V0f,V3x) != c_2Eextreal_2EPosInf ) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V1s) != c_2Eextreal_2EPosInf ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1e] :
( mem(V1e,A_27a)
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),c_2Epred__set_2EEMPTY(A_27a))) = ap(V0f,V1e) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__POS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V0f,V2x))) ) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V1s))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__SPOS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& V1s != c_2Epred__set_2EEMPTY(A_27a)
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V0f,V2x))) ) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V1s))) ) ) ) ) ).
fof(lameq_f3352,axiom,
! [A_27a,V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V4x] : ap(f3352(A_27a,V2f,V1P),V4x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(V1P,V4x)),ap(V2f,V4x)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IF__ELIM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(V1P,V3x)) ) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3352(A_27a,V2f,V1P)),V0s) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f),V0s) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__FINITE__SAME,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2p] :
( mem(V2p,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2p),V0s))
& ! [V3q] :
( mem(V3q,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3q),V0s))
=> ap(V1f,V2p) = ap(V1f,V3q) ) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))),ap(V1f,V2p)) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__FINITE__CONST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2x] :
( mem(V2x,ty_2Eextreal_2Eextreal)
=> ( ! [V3y] :
( mem(V3y,A_27a)
=> ap(V1f,V3y) = V2x )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0P) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0P))),V2x) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__ZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),k(A_27a,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))),V0s) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__0,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ap(V0f,V2x) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V1s) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(lameq_f3353,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V2x] : ap(f3353(A_27a,V1f,V0s),V2x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)),ap(V1f,V2x)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IN__IF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3353(A_27a,V1f,V0s)),V0s) ) ) ) ) ).
fof(lameq_f3354,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ! [V4x] : ap(f3354(A_27a,V1f,V2c),V4x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V2c)),ap(V1f,V4x)) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__CMUL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2c))
| ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x))) ) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3354(A_27a,V1f,V2c)),V0s) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V2c)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__CMUL2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> ap(V1f,V3x) != c_2Eextreal_2ENegInf ) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3354(A_27a,V1f,V2c)),V0s) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V2c)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f_27] :
( mem(V1f_27,arr(A_27a,A_27b))
=> ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V1f_27),V0s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f_27),V0s)))
=> ! [V2f] :
( mem(V2f,arr(A_27b,ty_2Eextreal_2Eextreal))
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),V2f),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f_27),V0s)) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Eextreal_2Eextreal,A_27b),V2f),V1f_27)),V0s) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__DISJOINT__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1s_27] :
( mem(V1s_27,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1s_27))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V1s_27)) )
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1s_27)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f),V0s)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f),V1s_27)) ) ) ) ) ) ).
fof(lameq_f3355,axiom,
! [A_27a,V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] : ap(f3355(A_27a,V0s),V1x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__EQ__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3355(A_27a,V0s)),V0s) = ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s)) ) ) ) ).
fof(lameq_f3356,axiom,
! [A_27a,V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] : ap(f3356(A_27a,V0s),V1x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)),ap(c_2Eextreal_2Eextreal__inv,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s)))),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__INV__CARD__EQ__1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ( V0s != c_2Epred__set_2EEMPTY(A_27a)
& p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3356(A_27a,V0s)),V0s) = ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ).
fof(lameq_f3357,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2p] : ap(f3357(A_27a,V1f),V2p) = ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V1f,V2p)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__INTER__NONZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),f3357(A_27a,V1f))) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__INTER__ELIM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s_27] :
( mem(V2s_27,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s_27))
=> ap(V1f,V3x) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V2s_27)) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__ZERO__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2t))
=> ap(V1f,V3x) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V2t)) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__MONO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V1f,V3x)),ap(V2f_27,V3x))) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f_27),V0s))) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__MONO__SET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V2t))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2t))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V0f,V3x))) ) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V1s)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V2t))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> ap(V1f,V3x) = ap(V2f_27,V3x) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f_27),V0s) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__POS__MEM__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V0f,V2x))) ) ) )
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V3x)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V1s))) ) ) ) ) ) ) ).
fof(lameq_f3358,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f3358(A_27a,V1f,V2f_27),V3x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V1f,V3x)),ap(V2f_27,V3x)) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3358(A_27a,V1f,V2f_27)),V0s) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f_27),V0s)) ) ) ) ) ) ).
fof(lameq_f3359,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5x] : ap(f3359(A_27a,V1f,V2f_27),V5x) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(V1f,V5x)),ap(V2f_27,V5x)) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__SUB,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> ap(V2f_27,V3x) != c_2Eextreal_2ENegInf ) )
| ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s))
=> ap(V2f_27,V4x) != c_2Eextreal_2EPosInf ) ) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3359(A_27a,V1f,V2f_27)),V0s) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f_27),V0s)) ) ) ) ) ) ) ).
fof(lameq_f3360,axiom,
! [A_27b,A_27a,V2f] :
( mem(V2f,arr(A_27a,arr(A_27b,ty_2Eextreal_2Eextreal)))
=> ! [V1s_27] :
( mem(V1s_27,arr(A_27b,bool))
=> ! [V3x] : ap(f3360(A_27b,A_27a,V2f,V1s_27),V3x) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),ap(V2f,V3x)),V1s_27) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__EXTREAL__SUM__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1s_27] :
( mem(V1s_27,arr(A_27b,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,arr(A_27b,ty_2Eextreal_2Eextreal)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V1s_27)) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3360(A_27b,A_27a,V2f,V1s_27)),V0s) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Epair_2Eprod(A_27a,A_27b)),f3092(A_27a,A_27b,V2f)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1s_27)) ) ) ) ) ) ) ).
fof(lameq_f3361,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x] : ap(f3361(A_27a,V0f),V2x) = ap(c_2Eextreal_2ENormal,ap(V0f,V2x)) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NORMAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3361(A_27a,V0f)),V1s) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V0f),V1s)) ) ) ) ) ).
fof(lameq_f3362,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ! [V3x] : ap(f3362(A_27a,V1f,V0s,V2z),V3x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)),ap(V1f,V3x)),V2z) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IN__IF__ALT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3362(A_27a,V1f,V0s,V2z)),V0s) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__CROSS__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Eextreal_2Eextreal))
=> ! [V1s1] :
( mem(V1s1,arr(A_27a,bool))
=> ! [V2s2] :
( mem(V2s2,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s1))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2s2)) )
=> ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Eextreal_2Eextreal),f3103(A_27b,A_27a,V0f))),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V1s1),V2s2)) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Epair_2Eprod(A_27b,A_27a)),ap(c_2Epair_2EUNCURRY(A_27b,A_27a,ty_2Eextreal_2Eextreal),f3105(A_27a,A_27b,V0f))),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27a),V2s2),V1s1)) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__COUNT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,c_2Enum_2E0)),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))
& ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,c_2Enum_2E0)),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))
& ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,c_2Enum_2E0)),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
& ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,c_2Enum_2E0)),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) ).
fof(lameq_f3363,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] : ap(f3363(V0p,V1x),V2y) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(V0p,V2y)),ap(ap(c_2Eextreal_2Eextreal__le,V2y),V1x)) ) ) ).
fof(lameq_f3364,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] : ap(f3364(V0p),V1x) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(ty_2Eextreal_2Eextreal),f3363(V0p,V1x))),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),V1x),c_2Eextreal_2EPosInf)) ) ).
fof(lameq_f3365,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V3x] : ap(f3365(V0p),V3x) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(V0p,V3x)),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),V3x),c_2Eextreal_2ENegInf)) ) ).
fof(lameq_f3366,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V4r] : ap(f3366(V0p),V4r) = ap(V0p,ap(c_2Eextreal_2ENormal,V4r)) ) ).
fof(ax_thm_2Eextreal_2Eextreal__sup__def,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ap(c_2Eextreal_2Eextreal__sup,V0p) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(c_2Ebool_2E_21(ty_2Eextreal_2Eextreal),f3364(V0p))),c_2Eextreal_2EPosInf),ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(c_2Ebool_2E_21(ty_2Eextreal_2Eextreal),f3365(V0p))),c_2Eextreal_2ENegInf),ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Esup,f3366(V0p))))) ) ).
fof(ax_thm_2Eextreal_2Eextreal__inf__def,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ap(c_2Eextreal_2Eextreal__inf,V0p) = ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),c_2Eextreal_2Eextreal__ainv),V0p))) ) ).
fof(conj_thm_2Eextreal_2Ele__sup__imp,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V1x))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),ap(c_2Eextreal_2Eextreal__sup,V0p))) ) ) ) ).
fof(conj_thm_2Eextreal_2Esup__le,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__sup,V0p)),V1x))
<=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V2y))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V2y),V1x)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__sup,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),ap(c_2Eextreal_2Eextreal__sup,V0p)))
<=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( ! [V3z] :
( mem(V3z,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V3z))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V3z),V2y)) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),V2y)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esup__eq,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( ap(c_2Eextreal_2Eextreal__sup,V0p) = V1x
<=> ( ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V2y))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V2y),V1x)) ) )
& ! [V3y] :
( mem(V3y,ty_2Eextreal_2Eextreal)
=> ( ! [V4z] :
( mem(V4z,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V4z))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V4z),V3y)) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),V3y)) ) ) ) ) ) ) ).
fof(lameq_f3367,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] : ap(f3367(V0x),V1y) = ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),V1y),V0x) ) ).
fof(conj_thm_2Eextreal_2Esup__const,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(c_2Eextreal_2Eextreal__sup,f3367(V0x)) = V0x ) ).
fof(conj_thm_2Eextreal_2Esup__const__alt,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1z] :
( mem(V1z,ty_2Eextreal_2Eextreal)
=> ( ( ? [V2x] :
( mem(V2x,ty_2Eextreal_2Eextreal)
& p(ap(V0p,V2x)) )
& ! [V3x] :
( mem(V3x,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V3x))
=> V3x = V1z ) ) )
=> ap(c_2Eextreal_2Eextreal__sup,V0p) = V1z ) ) ) ).
fof(conj_thm_2Eextreal_2Esup__const__over__set,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1k] :
( mem(V1k,ty_2Eextreal_2Eextreal)
=> ( V0s != c_2Epred__set_2EEMPTY(A_27a)
=> ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),k(A_27a,V1k)),V0s)) = V1k ) ) ) ) ).
fof(lameq_f3368,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1n] : ap(f3368(V0x),V1n) = ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),V0x),ap(c_2Eextreal_2Eextreal__of__num,V1n)) ) ).
fof(lameq_f3369,axiom,
! [V0x] : ap(f3369,V0x) = ap(c_2Ebool_2E_3F(ty_2Enum_2Enum),f3368(V0x)) ).
fof(conj_thm_2Eextreal_2Esup__num,axiom,
ap(c_2Eextreal_2Eextreal__sup,f3369) = c_2Eextreal_2EPosInf ).
fof(conj_thm_2Eextreal_2Esup__le__sup__imp,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1q] :
( mem(V1q,arr(ty_2Eextreal_2Eextreal,bool))
=> ( ! [V2x] :
( mem(V2x,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V2x))
=> ? [V3y] :
( mem(V3y,ty_2Eextreal_2Eextreal)
& p(ap(V1q,V3y))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V2x),V3y)) ) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__sup,V0p)),ap(c_2Eextreal_2Eextreal__sup,V1q))) ) ) ) ).
fof(conj_thm_2Eextreal_2Esup__mono,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1q] :
( mem(V1q,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0p,V2n)),ap(V1q,V2n))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V0p),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V1q),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) ) ) ) ).
fof(lameq_f3370,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V3n] : ap(f3370(V0f),V3n) = ap(V0f,ap(c_2Enum_2ESUC,V3n)) ) ).
fof(conj_thm_2Eextreal_2Esup__suc,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V1m)),ap(V0f,V2n))) ) ) )
=> ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3370(V0f)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ).
fof(lameq_f3371,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n] : ap(f3371(V0f),V2n) = ap(c_2Eextreal_2ENormal,ap(V0f,V2n)) ) ).
fof(conj_thm_2Eextreal_2Esup__seq,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ( p(ap(c_2Eextreal_2Emono__increasing,V0f))
=> ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V1l))
<=> ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3371(V0f)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(c_2Eextreal_2ENormal,V1l) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esup__lt__infty,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__sup,V0p)),c_2Eextreal_2EPosInf))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V1x))
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V1x),c_2Eextreal_2EPosInf)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esup__max,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1z] :
( mem(V1z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(V0p,V1z))
& ! [V2x] :
( mem(V2x,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V2x))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V2x),V1z)) ) ) )
=> ap(c_2Eextreal_2Eextreal__sup,V0p) = V1z ) ) ) ).
fof(lameq_f3372,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V6n] : ap(f3372(V0f,V1g),V6n) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,V6n)),ap(V1g,V6n)) ) ) ).
fof(conj_thm_2Eextreal_2Esup__add__mono,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V0f,V2n))) )
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V3n)),ap(V0f,ap(c_2Enum_2ESUC,V3n)))) )
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1g,V4n))) )
& ! [V5n] :
( mem(V5n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V1g,V5n)),ap(V1g,ap(c_2Enum_2ESUC,V5n)))) ) )
=> ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3372(V0f,V1g)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V1g),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) ) ) ) ).
fof(lameq_f3373,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal)))
=> ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> ! [V7i] : ap(f3373(V0f,V6n),V7i) = ap(ap(V0f,V7i),V6n) ) ) ).
fof(lameq_f3374,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal)))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ! [V6n] : ap(f3374(V0f,V1s),V6n) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f3373(V0f,V6n)),V1s) ) ) ).
fof(lameq_f3375,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal)))
=> ! [V8i] : ap(f3375(V0f),V8i) = ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),ap(V0f,V8i)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ).
fof(conj_thm_2Eextreal_2Esup__sum__mono,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal)))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s))
& ! [V2i] :
( mem(V2i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2i),V1s))
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(V0f,V2i),V3n))) ) ) )
& ! [V4i] :
( mem(V4i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4i),V1s))
=> ! [V5n] :
( mem(V5n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(V0f,V4i),V5n)),ap(ap(V0f,V4i),ap(c_2Enum_2ESUC,V5n)))) ) ) ) )
=> ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3374(V0f,V1s)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f3375(V0f)),V1s) ) ) ) ).
fof(conj_thm_2Eextreal_2Esup__le__mono,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1z] :
( mem(V1z,ty_2Eextreal_2Eextreal)
=> ( ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V2n)),ap(V0f,ap(c_2Enum_2ESUC,V2n)))) )
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1z),ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) )
=> ? [V3n] :
( mem(V3n,ty_2Enum_2Enum)
& p(ap(ap(c_2Eextreal_2Eextreal__le,V1z),ap(V0f,V3n))) ) ) ) ) ).
fof(lameq_f3376,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V2n] : ap(f3376(A_27a,V0f,V1c),V2n) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V1c)),ap(V0f,V2n)) ) ) ).
fof(conj_thm_2Eextreal_2Esup__cmul,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1c))
=> ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),f3376(A_27a,V0f,V1c)),c_2Epred__set_2EUNIV(A_27a))) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V1c)),ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V0f),c_2Epred__set_2EUNIV(A_27a)))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esup__lt,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ? [V2x] :
( mem(V2x,ty_2Eextreal_2Eextreal)
& p(ap(V0P,V2x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),V2x)) )
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,V1y),ap(c_2Eextreal_2Eextreal__sup,V0P))) ) ) ) ).
fof(conj_thm_2Eextreal_2Esup__lt__epsilon,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1e] :
( mem(V1e,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1e))
& ? [V2x] :
( mem(V2x,ty_2Eextreal_2Eextreal)
& p(ap(V0P,V2x))
& V2x != c_2Eextreal_2ENegInf )
& ap(c_2Eextreal_2Eextreal__sup,V0P) != c_2Eextreal_2EPosInf )
=> ? [V3x] :
( mem(V3x,ty_2Eextreal_2Eextreal)
& p(ap(V0P,V3x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__sup,V0P)),ap(ap(c_2Eextreal_2Eextreal__add,V3x),V1e))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Einf__le__imp,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V1x))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__inf,V0p)),V1x)) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__inf,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),ap(c_2Eextreal_2Eextreal__inf,V0p)))
<=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V2y))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),V2y)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Einf__le,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__inf,V0p)),V1x))
<=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( ! [V3z] :
( mem(V3z,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V3z))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V2y),V3z)) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V2y),V1x)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Einf__eq,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( ap(c_2Eextreal_2Eextreal__inf,V0p) = V1x
<=> ( ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V2y))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),V2y)) ) )
& ! [V3y] :
( mem(V3y,ty_2Eextreal_2Eextreal)
=> ( ! [V4z] :
( mem(V4z,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V4z))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V3y),V4z)) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V3y),V1x)) ) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Einf__const,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(c_2Eextreal_2Eextreal__inf,f3367(V0x)) = V0x ) ).
fof(conj_thm_2Eextreal_2Einf__const__alt,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1z] :
( mem(V1z,ty_2Eextreal_2Eextreal)
=> ( ( ? [V2x] :
( mem(V2x,ty_2Eextreal_2Eextreal)
& p(ap(V0p,V2x)) )
& ! [V3x] :
( mem(V3x,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V3x))
=> V3x = V1z ) ) )
=> ap(c_2Eextreal_2Eextreal__inf,V0p) = V1z ) ) ) ).
fof(conj_thm_2Eextreal_2Einf__const__over__set,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1k] :
( mem(V1k,ty_2Eextreal_2Eextreal)
=> ( V0s != c_2Epred__set_2EEMPTY(A_27a)
=> ap(c_2Eextreal_2Eextreal__inf,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),k(A_27a,V1k)),V0s)) = V1k ) ) ) ) ).
fof(conj_thm_2Eextreal_2Einf__suc,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V2n)),ap(V0f,V1m))) ) ) )
=> ap(c_2Eextreal_2Eextreal__inf,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3370(V0f)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(c_2Eextreal_2Eextreal__inf,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ).
fof(conj_thm_2Eextreal_2Einf__seq,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ( p(ap(c_2Eextreal_2Emono__decreasing,V0f))
=> ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V1l))
<=> ap(c_2Eextreal_2Eextreal__inf,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3371(V0f)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(c_2Eextreal_2ENormal,V1l) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Einf__lt__infty,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,c_2Eextreal_2ENegInf),ap(c_2Eextreal_2Eextreal__inf,V0p)))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V1x))
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,c_2Eextreal_2ENegInf),V1x)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Einf__min,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1z] :
( mem(V1z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(V0p,V1z))
& ! [V2x] :
( mem(V2x,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V2x))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1z),V2x)) ) ) )
=> ap(c_2Eextreal_2Eextreal__inf,V0p) = V1z ) ) ) ).
fof(lameq_f3377,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V2n] : ap(f3377(A_27a,V0f,V1c),V2n) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2ENormal,V1c)),ap(V0f,V2n)) ) ) ).
fof(conj_thm_2Eextreal_2Einf__cminus,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2ENormal,V1c)),ap(c_2Eextreal_2Eextreal__inf,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V0f),c_2Epred__set_2EUNIV(A_27a)))) = ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),f3377(A_27a,V0f,V1c)),c_2Epred__set_2EUNIV(A_27a))) ) ) ) ).
fof(lameq_f3378,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1n] : ap(f3378(V0f),V1n) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,V1n)) ) ).
fof(ax_thm_2Eextreal_2Eext__suminf__def,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ap(c_2Eextreal_2Eext__suminf,V0f) = ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3378(V0f)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ).
fof(lameq_f3379,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V3n] : ap(f3379(V0f,V1g),V3n) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,V3n)),ap(V1g,V3n)) ) ) ).
fof(conj_thm_2Eextreal_2Eext__suminf__add,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V0f,V2n)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1g,V2n))) ) )
=> ap(c_2Eextreal_2Eext__suminf,f3379(V0f,V1g)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2Eext__suminf,V0f)),ap(c_2Eextreal_2Eext__suminf,V1g)) ) ) ) ).
fof(lameq_f3380,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Eextreal_2Eextreal)
=> ! [V3n] : ap(f3380(V0f,V1c),V3n) = ap(ap(c_2Eextreal_2Eextreal__mul,V1c),ap(V0f,V3n)) ) ) ).
fof(conj_thm_2Eextreal_2Eext__suminf__cmul,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V1c))
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V0f,V2n))) ) )
=> ap(c_2Eextreal_2Eext__suminf,f3380(V0f,V1c)) = ap(ap(c_2Eextreal_2Eextreal__mul,V1c),ap(c_2Eextreal_2Eext__suminf,V0f)) ) ) ) ).
fof(lameq_f3381,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V4n] : ap(f3381(V0f,V1c),V4n) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V1c)),ap(V0f,V4n)) ) ) ).
fof(conj_thm_2Eextreal_2Eext__suminf__cmul__alt,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1c))
& ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(V0f,V2n) != c_2Eextreal_2ENegInf )
| ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(V0f,V3n) != c_2Eextreal_2EPosInf ) ) )
=> ap(c_2Eextreal_2Eext__suminf,f3381(V0f,V1c)) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V1c)),ap(c_2Eextreal_2Eext__suminf,V0f)) ) ) ) ).
fof(conj_thm_2Eextreal_2Eext__suminf__lt__infty,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ( ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V0f,V1n))) )
& ap(c_2Eextreal_2Eext__suminf,V0f) != c_2Eextreal_2EPosInf )
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V2n)),c_2Eextreal_2EPosInf)) ) ) ) ).
fof(lameq_f3382,axiom,
! [V0r] :
( mem(V0r,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n] : ap(f3382(V0r),V2n) = ap(c_2Eextreal_2ENormal,ap(V0r,V2n)) ) ).
fof(lameq_f3383,axiom,
! [V0r] :
( mem(V0r,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n] : ap(f3383(V0r),V3n) = ap(c_2Eextreal_2ENormal,ap(V0r,V3n)) ) ).
fof(conj_thm_2Eextreal_2Eext__suminf__suminf,axiom,
! [V0r] :
( mem(V0r,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0r,V1n))) )
& ap(c_2Eextreal_2Eext__suminf,f3382(V0r)) != c_2Eextreal_2EPosInf )
=> ap(c_2Eextreal_2Eext__suminf,f3383(V0r)) = ap(c_2Eextreal_2ENormal,ap(c_2Eseq_2Esuminf,V0r)) ) ) ).
fof(conj_thm_2Eextreal_2Eext__suminf__mono,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ap(V1g,V2n) != c_2Eextreal_2ENegInf
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V1g,V2n)),ap(V0f,V2n))) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eext__suminf,V1g)),ap(c_2Eextreal_2Eext__suminf,V0f))) ) ) ) ).
fof(lameq_f3384,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V3i] : ap(f3384(V0f,V1g),V3i) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(V0f,V3i)),ap(V1g,V3i)) ) ) ).
fof(conj_thm_2Eextreal_2Eext__suminf__sub,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1g,V2n)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V1g,V2n)),ap(V0f,V2n))) ) )
& ap(c_2Eextreal_2Eext__suminf,V0f) != c_2Eextreal_2EPosInf )
=> ap(c_2Eextreal_2Eext__suminf,f3384(V0f,V1g)) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2Eext__suminf,V0f)),ap(c_2Eextreal_2Eext__suminf,V1g)) ) ) ) ).
fof(conj_thm_2Eextreal_2Eext__suminf__sum,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V0f,V2n))) )
& ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),V3m))
=> ap(V0f,V3m) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) )
=> ap(c_2Eextreal_2Eext__suminf,V0f) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,V1n)) ) ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__min__def,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__min,V0x),V1y) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y)),V0x),V1y) ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__max__def,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__max,V0x),V1y) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y)),V1y),V0x) ) ) ).
fof(conj_thm_2Eextreal_2Emin__le,axiom,
! [V0z] :
( mem(V0z,ty_2Eextreal_2Eextreal)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__min,V1x),V2y)),V0z))
<=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),V0z))
| p(ap(ap(c_2Eextreal_2Eextreal__le,V2y),V0z)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Emin__le1,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__min,V0x),V1y)),V0x)) ) ) ).
fof(conj_thm_2Eextreal_2Emin__le2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__min,V0x),V1y)),V1y)) ) ) ).
fof(conj_thm_2Eextreal_2Ele__min,axiom,
! [V0z] :
( mem(V0z,ty_2Eextreal_2Eextreal)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0z),ap(ap(c_2Eextreal_2Eextreal__min,V1x),V2y)))
<=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0z),V1x))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V0z),V2y)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Emin__le2__imp,axiom,
! [V0x1] :
( mem(V0x1,ty_2Eextreal_2Eextreal)
=> ! [V1x2] :
( mem(V1x2,ty_2Eextreal_2Eextreal)
=> ! [V2y1] :
( mem(V2y1,ty_2Eextreal_2Eextreal)
=> ! [V3y2] :
( mem(V3y2,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0x1),V2y1))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V1x2),V3y2)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__min,V0x1),V1x2)),ap(ap(c_2Eextreal_2Eextreal__min,V2y1),V3y2))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Emin__refl,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__min,V0x),V0x) = V0x ) ).
fof(conj_thm_2Eextreal_2Emin__comm,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__min,V0x),V1y) = ap(ap(c_2Eextreal_2Eextreal__min,V1y),V0x) ) ) ).
fof(conj_thm_2Eextreal_2Emin__infty,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( ap(ap(c_2Eextreal_2Eextreal__min,V0x),c_2Eextreal_2EPosInf) = V0x
& ap(ap(c_2Eextreal_2Eextreal__min,c_2Eextreal_2EPosInf),V0x) = V0x
& ap(ap(c_2Eextreal_2Eextreal__min,c_2Eextreal_2ENegInf),V0x) = c_2Eextreal_2ENegInf
& ap(ap(c_2Eextreal_2Eextreal__min,V0x),c_2Eextreal_2ENegInf) = c_2Eextreal_2ENegInf ) ) ).
fof(conj_thm_2Eextreal_2Ele__max,axiom,
! [V0z] :
( mem(V0z,ty_2Eextreal_2Eextreal)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0z),ap(ap(c_2Eextreal_2Eextreal__max,V1x),V2y)))
<=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0z),V1x))
| p(ap(ap(c_2Eextreal_2Eextreal__le,V0z),V2y)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__max1,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),ap(ap(c_2Eextreal_2Eextreal__max,V0x),V1y))) ) ) ).
fof(conj_thm_2Eextreal_2Ele__max2,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(ap(c_2Eextreal_2Eextreal__max,V0x),V1y))) ) ) ).
fof(conj_thm_2Eextreal_2Emax__le,axiom,
! [V0z] :
( mem(V0z,ty_2Eextreal_2Eextreal)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__max,V1x),V2y)),V0z))
<=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),V0z))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V2y),V0z)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Emax__le2__imp,axiom,
! [V0x1] :
( mem(V0x1,ty_2Eextreal_2Eextreal)
=> ! [V1x2] :
( mem(V1x2,ty_2Eextreal_2Eextreal)
=> ! [V2y1] :
( mem(V2y1,ty_2Eextreal_2Eextreal)
=> ! [V3y2] :
( mem(V3y2,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,V0x1),V2y1))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V1x2),V3y2)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__max,V0x1),V1x2)),ap(ap(c_2Eextreal_2Eextreal__max,V2y1),V3y2))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Emax__refl,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__max,V0x),V0x) = V0x ) ).
fof(conj_thm_2Eextreal_2Emax__comm,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__max,V0x),V1y) = ap(ap(c_2Eextreal_2Eextreal__max,V1y),V0x) ) ) ).
fof(conj_thm_2Eextreal_2Emax__infty,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( ap(ap(c_2Eextreal_2Eextreal__max,V0x),c_2Eextreal_2EPosInf) = c_2Eextreal_2EPosInf
& ap(ap(c_2Eextreal_2Eextreal__max,c_2Eextreal_2EPosInf),V0x) = c_2Eextreal_2EPosInf
& ap(ap(c_2Eextreal_2Eextreal__max,c_2Eextreal_2ENegInf),V0x) = V0x
& ap(ap(c_2Eextreal_2Eextreal__max,V0x),c_2Eextreal_2ENegInf) = V0x ) ) ).
fof(lameq_f3385,axiom,
! [V1a] :
( mem(V1a,ty_2Enum_2Enum)
=> ! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V2b] : ap(f3385(V1a,V0x),V2b) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),V0x),ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,V1a)),ap(c_2Eextreal_2Eextreal__of__num,V2b)))),ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(c_2Eextreal_2Eextreal__of__num,V2b))) ) ) ).
fof(lameq_f3386,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1a] : ap(f3386(V0x),V1a) = ap(c_2Ebool_2E_3F(ty_2Enum_2Enum),f3385(V1a,V0x)) ) ).
fof(lameq_f3387,axiom,
! [V0x] : ap(f3387,V0x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V0x),ap(c_2Ebool_2E_3F(ty_2Enum_2Enum),f3386(V0x))) ).
fof(lameq_f3388,axiom,
! [V4a] :
( mem(V4a,ty_2Enum_2Enum)
=> ! [V3x] :
( mem(V3x,ty_2Eextreal_2Eextreal)
=> ! [V5b] : ap(f3388(V4a,V3x),V5b) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),V3x),ap(c_2Eextreal_2Eextreal__ainv,ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,V4a)),ap(c_2Eextreal_2Eextreal__of__num,V5b))))),ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(c_2Eextreal_2Eextreal__of__num,V5b))) ) ) ).
fof(lameq_f3389,axiom,
! [V3x] :
( mem(V3x,ty_2Eextreal_2Eextreal)
=> ! [V4a] : ap(f3389(V3x),V4a) = ap(c_2Ebool_2E_3F(ty_2Enum_2Enum),f3388(V4a,V3x)) ) ).
fof(lameq_f3390,axiom,
! [V3x] : ap(f3390,V3x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V3x),ap(c_2Ebool_2E_3F(ty_2Enum_2Enum),f3389(V3x))) ).
fof(ax_thm_2Eextreal_2EQ__set__def,axiom,
c_2Eextreal_2EQ__set = ap(ap(c_2Epred__set_2EUNION(ty_2Eextreal_2Eextreal),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3387)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3390)) ).
fof(conj_thm_2Eextreal_2EQ__not__infty,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V0x),c_2Eextreal_2EQ__set))
=> ? [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
& V0x = ap(c_2Eextreal_2ENormal,V1y) ) ) ) ).
fof(conj_thm_2Eextreal_2EQ__COUNTABLE,axiom,
p(ap(c_2Epred__set_2Ecountable(ty_2Eextreal_2Eextreal),c_2Eextreal_2EQ__set)) ).
fof(conj_thm_2Eextreal_2ENUM__IN__Q,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(c_2Eextreal_2Eextreal__of__num,V0n)),c_2Eextreal_2EQ__set))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2Eextreal__of__num,V0n))),c_2Eextreal_2EQ__set)) ) ) ).
fof(conj_thm_2Eextreal_2EQ__INFINITE,axiom,
~ p(ap(c_2Epred__set_2EFINITE(ty_2Eextreal_2Eextreal),c_2Eextreal_2EQ__set)) ).
fof(conj_thm_2Eextreal_2EOPP__IN__Q,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V0x),c_2Eextreal_2EQ__set))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(c_2Eextreal_2Eextreal__ainv,V0x)),c_2Eextreal_2EQ__set)) ) ) ).
fof(conj_thm_2Eextreal_2EINV__IN__Q,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V0x),c_2Eextreal_2EQ__set))
& V0x != ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x)),c_2Eextreal_2EQ__set)) ) ) ).
fof(conj_thm_2Eextreal_2ECMUL__IN__Q,axiom,
! [V0z] :
( mem(V0z,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V1x),c_2Eextreal_2EQ__set))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__of__num,V0z)),V1x)),c_2Eextreal_2EQ__set))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2Eextreal__of__num,V0z))),V1x)),c_2Eextreal_2EQ__set)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EADD__IN__Q,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V0x),c_2Eextreal_2EQ__set))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V1y),c_2Eextreal_2EQ__set)) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__add,V0x),V1y)),c_2Eextreal_2EQ__set)) ) ) ) ).
fof(conj_thm_2Eextreal_2ESUB__IN__Q,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V0x),c_2Eextreal_2EQ__set))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V1y),c_2Eextreal_2EQ__set)) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__sub,V0x),V1y)),c_2Eextreal_2EQ__set)) ) ) ) ).
fof(conj_thm_2Eextreal_2EMUL__IN__Q,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V0x),c_2Eextreal_2EQ__set))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V1y),c_2Eextreal_2EQ__set)) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y)),c_2Eextreal_2EQ__set)) ) ) ) ).
fof(conj_thm_2Eextreal_2EDIV__IN__Q,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V0x),c_2Eextreal_2EQ__set))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V1y),c_2Eextreal_2EQ__set))
& V1y != ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__div,V0x),V1y)),c_2Eextreal_2EQ__set)) ) ) ) ).
fof(conj_thm_2Eextreal_2Erat__not__infty,axiom,
! [V0r] :
( mem(V0r,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V0r),c_2Eextreal_2EQ__set))
=> ( V0r != c_2Eextreal_2ENegInf
& V0r != c_2Eextreal_2EPosInf ) ) ) ).
fof(ax_thm_2Eextreal_2Eceiling__def,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Eextreal_2Eceiling,ap(c_2Eextreal_2ENormal,V0x)) = ap(c_2Ewhile_2ELEAST,f2739(V0x)) ) ).
fof(conj_thm_2Eextreal_2ECEILING__LBOUND,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2ENormal,V0x)),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Eextreal_2Eceiling,ap(c_2Eextreal_2ENormal,V0x))))) ) ).
fof(conj_thm_2Eextreal_2ECEILING__UBOUND,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x))
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Eextreal_2Eceiling,ap(c_2Eextreal_2ENormal,V0x)))),ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2ENormal,V0x)),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))) ) ) ).
fof(conj_thm_2Eextreal_2EQ__DENSE__IN__R__LEMMA,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y)) )
=> ? [V2r] :
( mem(V2r,ty_2Eextreal_2Eextreal)
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V2r),c_2Eextreal_2EQ__set))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V2r))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V2r),V1y)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2EQ__DENSE__IN__R,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V1y))
=> ? [V2r] :
( mem(V2r,ty_2Eextreal_2Eextreal)
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V2r),c_2Eextreal_2EQ__set))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V0x),V2r))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,V2r),V1y)) ) ) ) ) ).
fof(conj_thm_2Eextreal_2ECOUNTABLE__ENUM__Q,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0c] :
( mem(V0c,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0c))
<=> ( V0c = c_2Epred__set_2EEMPTY(A_27a)
| ? [V1f] :
( mem(V1f,arr(ty_2Eextreal_2Eextreal,A_27a))
& V0c = ap(ap(c_2Epred__set_2EIMAGE(ty_2Eextreal_2Eextreal,A_27a),V1f),c_2Eextreal_2EQ__set) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2ECROSS__COUNTABLE__UNIV,axiom,
p(ap(c_2Epred__set_2Ecountable(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) ).
fof(conj_thm_2Eextreal_2ECROSS__COUNTABLE__LEMMA1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0t] :
( mem(V0t,arr(A_27b,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2Ecountable(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2Ecountable(A_27b),V0t)) )
=> p(ap(c_2Epred__set_2Ecountable(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V1s),V0t))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2ECROSS__COUNTABLE__LEMMA2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0t] :
( mem(V0t,arr(A_27b,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2Ecountable(A_27a),V1s))
& p(ap(c_2Epred__set_2Ecountable(A_27b),V0t))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V0t)) )
=> p(ap(c_2Epred__set_2Ecountable(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V1s),V0t))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2ECROSS__COUNTABLE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0t] :
( mem(V0t,arr(A_27b,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2Ecountable(A_27a),V1s))
& p(ap(c_2Epred__set_2Ecountable(A_27b),V0t)) )
=> p(ap(c_2Epred__set_2Ecountable(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V1s),V0t))) ) ) ) ) ) ).
fof(lameq_f3391,axiom,
! [V1b] :
( mem(V1b,ty_2Eextreal_2Eextreal)
=> ! [V0a] :
( mem(V0a,ty_2Eextreal_2Eextreal)
=> ! [V2x] : ap(f3391(V1b,V0a),V2x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V2x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,V0a),V2x)),ap(ap(c_2Eextreal_2Eextreal__lt,V2x),V1b))) ) ) ).
fof(ax_thm_2Eextreal_2Eopen__interval__def,axiom,
! [V0a] :
( mem(V0a,ty_2Eextreal_2Eextreal)
=> ! [V1b] :
( mem(V1b,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eopen__interval,V0a),V1b) = ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3391(V1b,V0a)) ) ) ).
fof(lameq_f3392,axiom,
! [V0a] :
( mem(V0a,ty_2Eextreal_2Eextreal)
=> ! [V1b] : ap(f3392(V0a),V1b) = ap(ap(c_2Epair_2E_2C(arr(ty_2Eextreal_2Eextreal,bool),bool),ap(ap(c_2Eextreal_2Eopen__interval,V0a),V1b)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V0a),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))),ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V1b),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) ) ).
fof(lameq_f3393,axiom,
! [V0a] : ap(f3393,V0a) = f3392(V0a) ).
fof(ax_thm_2Eextreal_2Eopen__intervals__set__def,axiom,
c_2Eextreal_2Eopen__intervals__set = ap(c_2Epred__set_2EGSPEC(arr(ty_2Eextreal_2Eextreal,bool),ty_2Epair_2Eprod(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)),ap(c_2Epair_2EUNCURRY(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal,ty_2Epair_2Eprod(arr(ty_2Eextreal_2Eextreal,bool),bool)),f3393)) ).
fof(lameq_f3394,axiom,
! [V0a] :
( mem(V0a,ty_2Eextreal_2Eextreal)
=> ! [V1b] : ap(f3394(V0a),V1b) = ap(ap(c_2Epair_2E_2C(arr(ty_2Eextreal_2Eextreal,bool),bool),ap(ap(c_2Eextreal_2Eopen__interval,V0a),V1b)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V0a),c_2Eextreal_2EQ__set)),ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V1b),c_2Eextreal_2EQ__set))) ) ).
fof(lameq_f3395,axiom,
! [V0a] : ap(f3395,V0a) = f3394(V0a) ).
fof(ax_thm_2Eextreal_2Erational__intervals__def,axiom,
c_2Eextreal_2Erational__intervals = ap(c_2Epred__set_2EGSPEC(arr(ty_2Eextreal_2Eextreal,bool),ty_2Epair_2Eprod(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)),ap(c_2Epair_2EUNCURRY(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal,ty_2Epair_2Eprod(arr(ty_2Eextreal_2Eextreal,bool),bool)),f3395)) ).
fof(conj_thm_2Eextreal_2ECOUNTABLE__RATIONAL__INTERVALS,axiom,
p(ap(c_2Epred__set_2Ecountable(arr(ty_2Eextreal_2Eextreal,bool)),c_2Eextreal_2Erational__intervals)) ).
%------------------------------------------------------------------------------