ITP001 Axioms: ITP124_5.ax
%------------------------------------------------------------------------------
% File : ITP124_5 : TPTP v8.2.0. 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 : nets_2.ax [Gau20]
% : HL4124_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 64 ( 7 unt; 18 typ; 0 def)
% Number of atoms : 1597 ( 19 equ)
% Maximal formula atoms : 20 ( 24 avg)
% Number of connectives : 217 ( 5 ~; 0 |; 43 &)
% ( 11 <=>; 158 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 10 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 1339 (1339 fml; 0 var)
% Number of types : 2 ( 1 usr)
% Number of type conns : 38 ( 17 >; 21 *; 0 +; 0 <<)
% Number of predicates : 36 ( 35 usr; 19 prp; 0-2 aty)
% Number of functors : 17 ( 17 usr; 0 con; 1-3 aty)
% Number of variables : 215 ( 199 !; 16 ?; 215 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Enets_2Ebounded,type,
c_2Enets_2Ebounded: ( del * del ) > $i ).
tff(mem_c_2Enets_2Ebounded,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Enets_2Ebounded(A_27a,A_27b),arr(ty_2Epair_2Eprod(ty_2Emetric_2Emetric(A_27a),arr(A_27b,arr(A_27b,bool))),arr(arr(A_27b,A_27a),bool))) ).
tff(tp_c_2Enets_2Edorder,type,
c_2Enets_2Edorder: del > $i ).
tff(mem_c_2Enets_2Edorder,axiom,
! [A_27a: del] : mem(c_2Enets_2Edorder(A_27a),arr(arr(A_27a,arr(A_27a,bool)),bool)) ).
tff(tp_c_2Enets_2Etends,type,
c_2Enets_2Etends: ( del * del ) > $i ).
tff(mem_c_2Enets_2Etends,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Enets_2Etends(A_27a,A_27b),arr(arr(A_27b,A_27a),arr(A_27a,arr(ty_2Epair_2Eprod(ty_2Etopology_2Etopology(A_27a),arr(A_27b,arr(A_27b,bool))),bool)))) ).
tff(tp_c_2Enets_2Etendsto,type,
c_2Enets_2Etendsto: del > $i ).
tff(mem_c_2Enets_2Etendsto,axiom,
! [A_27a: del] : mem(c_2Enets_2Etendsto(A_27a),arr(ty_2Epair_2Eprod(ty_2Emetric_2Emetric(A_27a),A_27a),arr(A_27a,arr(A_27a,bool)))) ).
tff(ax_thm_2Enets_2Edorder,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Enets_2Edorder(A_27a),V0g))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( ( p(ap(ap(V0g,V1x),V1x))
& p(ap(ap(V0g,V2y),V2y)) )
=> ? [V3z: $i] :
( mem(V3z,A_27a)
& p(ap(ap(V0g,V3z),V3z))
& ! [V4w: $i] :
( mem(V4w,A_27a)
=> ( p(ap(ap(V0g,V4w),V3z))
=> ( p(ap(ap(V0g,V4w),V1x))
& p(ap(ap(V0g,V4w),V2y)) ) ) ) ) ) ) ) ) ) ).
tff(ax_thm_2Enets_2Etends,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,arr(A_27b,A_27a))
=> ! [V1l: $i] :
( mem(V1l,A_27a)
=> ! [V2top: $i] :
( mem(V2top,ty_2Etopology_2Etopology(A_27a))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27b,arr(A_27b,bool)))
=> ( p(ap(ap(ap(c_2Enets_2Etends(A_27a,A_27b),V0s),V1l),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(A_27a),arr(A_27b,arr(A_27b,bool))),V2top),V3g)))
<=> ! [V4N: $i] :
( mem(V4N,arr(A_27a,bool))
=> ( p(ap(ap(c_2Etopology_2Eneigh(A_27a),V2top),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),A_27a),V4N),V1l)))
=> ? [V5n: $i] :
( mem(V5n,A_27b)
& p(ap(ap(V3g,V5n),V5n))
& ! [V6m: $i] :
( mem(V6m,A_27b)
=> ( p(ap(ap(V3g,V6m),V5n))
=> p(ap(V4N,ap(V0s,V6m))) ) ) ) ) ) ) ) ) ) ) ).
tff(ax_thm_2Enets_2Ebounded,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( mem(V0m,ty_2Emetric_2Emetric(A_27a))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27b,arr(A_27b,bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27b,A_27a))
=> ( p(ap(ap(c_2Enets_2Ebounded(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(ty_2Emetric_2Emetric(A_27a),arr(A_27b,arr(A_27b,bool))),V0m),V1g)),V2f))
<=> ? [V3k: tp__ty_2Erealax_2Ereal,V4x: $i] :
( mem(V4x,A_27a)
& ? [V5N: $i] :
( mem(V5N,A_27b)
& p(ap(ap(V1g,V5N),V5N))
& ! [V6n: $i] :
( mem(V6n,A_27b)
=> ( p(ap(ap(V1g,V6n),V5N))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),ap(V2f,V6n)),V4x))),inj__ty_2Erealax_2Ereal(V3k))) ) ) ) ) ) ) ) ) ).
tff(ax_thm_2Enets_2Etendsto,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Emetric_2Emetric(A_27a))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ! [V3z: $i] :
( mem(V3z,A_27a)
=> ( p(ap(ap(ap(c_2Enets_2Etendsto(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Emetric_2Emetric(A_27a),A_27a),V0m),V1x)),V2y),V3z))
<=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y))))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y))),ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V3z)))) ) ) ) ) ) ) ).
tff(conj_thm_2Enets_2EDORDER__LEMMA,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Enets_2Edorder(A_27a),V0g))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2Q: $i] :
( mem(V2Q,arr(A_27a,bool))
=> ( ( ? [V3n: $i] :
( mem(V3n,A_27a)
& p(ap(ap(V0g,V3n),V3n))
& ! [V4m: $i] :
( mem(V4m,A_27a)
=> ( p(ap(ap(V0g,V4m),V3n))
=> p(ap(V1P,V4m)) ) ) )
& ? [V5n: $i] :
( mem(V5n,A_27a)
& p(ap(ap(V0g,V5n),V5n))
& ! [V6m: $i] :
( mem(V6m,A_27a)
=> ( p(ap(ap(V0g,V6m),V5n))
=> p(ap(V2Q,V6m)) ) ) ) )
=> ? [V7n: $i] :
( mem(V7n,A_27a)
& p(ap(ap(V0g,V7n),V7n))
& ! [V8m: $i] :
( mem(V8m,A_27a)
=> ( p(ap(ap(V0g,V8m),V7n))
=> ( p(ap(V1P,V8m))
& p(ap(V2Q,V8m)) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Enets_2EDORDER__NGE,axiom,
p(ap(c_2Enets_2Edorder(ty_2Enum_2Enum),c_2Earithmetic_2E_3E_3D)) ).
tff(conj_thm_2Enets_2EDORDER__TENDSTO,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Emetric_2Emetric(A_27a))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(ap(c_2Enets_2Edorder(A_27a),ap(c_2Enets_2Etendsto(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Emetric_2Emetric(A_27a),A_27a),V0m),V1x)))) ) ) ).
tff(conj_thm_2Enets_2EMTOP__TENDS,axiom,
! [A_27a: del,A_27b: del,V0d: $i] :
( mem(V0d,ty_2Emetric_2Emetric(A_27a))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27b,arr(A_27b,bool)))
=> ! [V2x: $i] :
( mem(V2x,arr(A_27b,A_27a))
=> ! [V3x0: $i] :
( mem(V3x0,A_27a)
=> ( p(ap(ap(ap(c_2Enets_2Etends(A_27a,A_27b),V2x),V3x0),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(A_27a),arr(A_27b,arr(A_27b,bool))),ap(c_2Emetric_2Emtop(A_27a),V0d)),V1g)))
<=> ! [V4e: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4e)))
=> ? [V5n: $i] :
( mem(V5n,A_27b)
& p(ap(ap(V1g,V5n),V5n))
& ! [V6m: $i] :
( mem(V6m,A_27b)
=> ( p(ap(ap(V1g,V6m),V5n))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Emetric_2Edist(A_27a),V0d),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),ap(V2x,V6m)),V3x0))),inj__ty_2Erealax_2Ereal(V4e))) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Enets_2EMTOP__TENDS__UNIQ,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,arr(A_27b,A_27a))
=> ! [V1x0: $i] :
( mem(V1x0,A_27a)
=> ! [V2x1: $i] :
( mem(V2x1,A_27a)
=> ! [V3g: $i] :
( mem(V3g,arr(A_27b,arr(A_27b,bool)))
=> ! [V4d: $i] :
( mem(V4d,ty_2Emetric_2Emetric(A_27a))
=> ( p(ap(c_2Enets_2Edorder(A_27b),V3g))
=> ( ( p(ap(ap(ap(c_2Enets_2Etends(A_27a,A_27b),V0x),V1x0),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(A_27a),arr(A_27b,arr(A_27b,bool))),ap(c_2Emetric_2Emtop(A_27a),V4d)),V3g)))
& p(ap(ap(ap(c_2Enets_2Etends(A_27a,A_27b),V0x),V2x1),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(A_27a),arr(A_27b,arr(A_27b,bool))),ap(c_2Emetric_2Emtop(A_27a),V4d)),V3g))) )
=> ( V1x0 = V2x1 ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Enets_2ESEQ__TENDS,axiom,
! [A_27a: del,V0d: $i] :
( mem(V0d,ty_2Emetric_2Emetric(A_27a))
=> ! [V1x: $i] :
( mem(V1x,arr(ty_2Enum_2Enum,A_27a))
=> ! [V2x0: $i] :
( mem(V2x0,A_27a)
=> ( p(ap(ap(ap(c_2Enets_2Etends(A_27a,ty_2Enum_2Enum),V1x),V2x0),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(A_27a),arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))),ap(c_2Emetric_2Emtop(A_27a),V0d)),c_2Earithmetic_2E_3E_3D)))
<=> ! [V3e: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e)))
=> ? [V4N: tp__ty_2Enum_2Enum] :
! [V5n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3E_3D,inj__ty_2Enum_2Enum(V5n)),inj__ty_2Enum_2Enum(V4N)))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Emetric_2Edist(A_27a),V0d),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),ap(V1x,inj__ty_2Enum_2Enum(V5n))),V2x0))),inj__ty_2Erealax_2Ereal(V3e))) ) ) ) ) ) ) ).
tff(conj_thm_2Enets_2ELIM__TENDS,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( mem(V0m1,ty_2Emetric_2Emetric(A_27a))
=> ! [V1m2: $i] :
( mem(V1m2,ty_2Emetric_2Emetric(A_27b))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3x0: $i] :
( mem(V3x0,A_27a)
=> ! [V4y0: $i] :
( mem(V4y0,A_27b)
=> ( p(ap(ap(ap(c_2Etopology_2Elimpt(A_27a),ap(c_2Emetric_2Emtop(A_27a),V0m1)),V3x0),c_2Epred__set_2EUNIV(A_27a)))
=> ( p(ap(ap(ap(c_2Enets_2Etends(A_27b,A_27a),V2f),V4y0),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(A_27b),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(A_27b),V1m2)),ap(c_2Enets_2Etendsto(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Emetric_2Emetric(A_27a),A_27a),V0m1),V3x0)))))
<=> ! [V5e: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V5e)))
=> ? [V6d: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V6d)))
& ! [V7x: $i] :
( mem(V7x,A_27a)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Emetric_2Edist(A_27a),V0m1),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V7x),V3x0))))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emetric_2Edist(A_27a),V0m1),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V7x),V3x0))),inj__ty_2Erealax_2Ereal(V6d))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Emetric_2Edist(A_27b),V1m2),ap(ap(c_2Epair_2E_2C(A_27b,A_27b),ap(V2f,V7x)),V4y0))),inj__ty_2Erealax_2Ereal(V5e))) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Enets_2ELIM__TENDS2,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( mem(V0m1,ty_2Emetric_2Emetric(A_27a))
=> ! [V1m2: $i] :
( mem(V1m2,ty_2Emetric_2Emetric(A_27b))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3x0: $i] :
( mem(V3x0,A_27a)
=> ! [V4y0: $i] :
( mem(V4y0,A_27b)
=> ( p(ap(ap(ap(c_2Etopology_2Elimpt(A_27a),ap(c_2Emetric_2Emtop(A_27a),V0m1)),V3x0),c_2Epred__set_2EUNIV(A_27a)))
=> ( p(ap(ap(ap(c_2Enets_2Etends(A_27b,A_27a),V2f),V4y0),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(A_27b),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(A_27b),V1m2)),ap(c_2Enets_2Etendsto(A_27a),ap(ap(c_2Epair_2E_2C(ty_2Emetric_2Emetric(A_27a),A_27a),V0m1),V3x0)))))
<=> ! [V5e: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V5e)))
=> ? [V6d: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V6d)))
& ! [V7x: $i] :
( mem(V7x,A_27a)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Emetric_2Edist(A_27a),V0m1),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V7x),V3x0))))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Emetric_2Edist(A_27a),V0m1),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V7x),V3x0))),inj__ty_2Erealax_2Ereal(V6d))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Emetric_2Edist(A_27b),V1m2),ap(ap(c_2Epair_2E_2C(A_27b,A_27b),ap(V2f,V7x)),V4y0))),inj__ty_2Erealax_2Ereal(V5e))) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Enets_2EMR1__BOUNDED,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( p(ap(ap(c_2Enets_2Ebounded(ty_2Erealax_2Ereal,A_27a),ap(ap(c_2Epair_2E_2C(ty_2Emetric_2Emetric(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),c_2Emetric_2Emr1),V0g)),V1f))
<=> ? [V2k: tp__ty_2Erealax_2Ereal,V3N: $i] :
( mem(V3N,A_27a)
& p(ap(ap(V0g,V3N),V3N))
& ! [V4n: $i] :
( mem(V4n,A_27a)
=> ( p(ap(ap(V0g,V4n),V3N))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(V1f,V4n))),inj__ty_2Erealax_2Ereal(V2k))) ) ) ) ) ) ) ).
tff(stp_c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal,type,
tp__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal: $tType ).
tff(stp_inj_c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal,type,
inj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal: tp__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal > $i ).
tff(stp_surj_c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal,type,
surj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal: $i > tp__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal ).
tff(stp_inj_surj_c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal] : ( surj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal(inj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal(X)) = X ) ).
tff(stp_inj_mem_c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal] : mem(inj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal(X),ty_2Etopology_2Etopology(ty_2Erealax_2Ereal)) ).
tff(stp_iso_mem_c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal,axiom,
! [X: $i] :
( mem(X,ty_2Etopology_2Etopology(ty_2Erealax_2Ereal))
=> ( X = inj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal(surj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal(X)) ) ) ).
tff(lamtp_f2753,type,
f2753: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f2753,axiom,
! [A_27a: del,V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x0: tp__ty_2Erealax_2Ereal,V3n: $i] : ( ap(f2753(A_27a,V1x,V2x0),V3n) = ap(ap(c_2Ereal_2Ereal__sub,ap(V1x,V3n)),inj__ty_2Erealax_2Ereal(V2x0)) ) ) ).
tff(conj_thm_2Enets_2ENET__NULL,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),inj__ty_2Erealax_2Ereal(V2x0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
<=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),f2753(A_27a,V1x,V2x0)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) ) ) ) ).
tff(conj_thm_2Enets_2ENET__CONV__BOUNDED,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),inj__ty_2Erealax_2Ereal(V2x0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
=> p(ap(ap(c_2Enets_2Ebounded(ty_2Erealax_2Ereal,A_27a),ap(ap(c_2Epair_2E_2C(ty_2Emetric_2Emetric(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),c_2Emetric_2Emr1),V0g)),V1x)) ) ) ) ).
tff(conj_thm_2Enets_2ENET__CONV__NZ,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),inj__ty_2Erealax_2Ereal(V2x0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
& ( V2x0 != surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) )
=> ? [V3N: $i] :
( mem(V3N,A_27a)
& p(ap(ap(V0g,V3N),V3N))
& ! [V4n: $i] :
( mem(V4n,A_27a)
=> ( p(ap(ap(V0g,V4n),V3N))
=> ( surj__ty_2Erealax_2Ereal(ap(V1x,V4n)) != surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ) ) ) ).
tff(lamtp_f2754,type,
f2754: ( del * $i ) > $i ).
tff(lameq_f2754,axiom,
! [A_27a: del,V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3n: $i] : ( ap(f2754(A_27a,V1x),V3n) = ap(c_2Erealax_2Einv,ap(V1x,V3n)) ) ) ).
tff(conj_thm_2Enets_2ENET__CONV__IBOUNDED,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),inj__ty_2Erealax_2Ereal(V2x0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
& ( V2x0 != surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) )
=> p(ap(ap(c_2Enets_2Ebounded(ty_2Erealax_2Ereal,A_27a),ap(ap(c_2Epair_2E_2C(ty_2Emetric_2Emetric(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),c_2Emetric_2Emr1),V0g)),f2754(A_27a,V1x))) ) ) ) ).
tff(lamtp_f2755,type,
f2755: ( del * $i * $i ) > $i ).
tff(lameq_f2755,axiom,
! [A_27a: del,V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2y: $i] :
( mem(V2y,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3n: $i] : ( ap(f2755(A_27a,V1x,V2y),V3n) = ap(ap(c_2Erealax_2Ereal__add,ap(V1x,V3n)),ap(V2y,V3n)) ) ) ) ).
tff(conj_thm_2Enets_2ENET__NULL__ADD,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Enets_2Edorder(A_27a),V0g))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2y: $i] :
( mem(V2y,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
& p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V2y),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) )
=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),f2755(A_27a,V1x,V2y)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) ) ) ) ) ) ).
tff(lamtp_f2756,type,
f2756: ( del * $i * $i ) > $i ).
tff(lameq_f2756,axiom,
! [A_27a: del,V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2y: $i] :
( mem(V2y,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3n: $i] : ( ap(f2756(A_27a,V1x,V2y),V3n) = ap(ap(c_2Erealax_2Ereal__mul,ap(V1x,V3n)),ap(V2y,V3n)) ) ) ) ).
tff(conj_thm_2Enets_2ENET__NULL__MUL,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Enets_2Edorder(A_27a),V0g))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2y: $i] :
( mem(V2y,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ( p(ap(ap(c_2Enets_2Ebounded(ty_2Erealax_2Ereal,A_27a),ap(ap(c_2Epair_2E_2C(ty_2Emetric_2Emetric(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),c_2Emetric_2Emr1),V0g)),V1x))
& p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V2y),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) )
=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),f2756(A_27a,V1x,V2y)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) ) ) ) ) ) ).
tff(lamtp_f2757,type,
f2757: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f2757,axiom,
! [A_27a: del,V2x: $i] :
( mem(V2x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1k: tp__ty_2Erealax_2Ereal,V3n: $i] : ( ap(f2757(A_27a,V2x,V1k),V3n) = ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1k)),ap(V2x,V3n)) ) ) ).
tff(conj_thm_2Enets_2ENET__NULL__CMUL,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ! [V1k: tp__ty_2Erealax_2Ereal,V2x: $i] :
( mem(V2x,arr(A_27a,ty_2Erealax_2Ereal))
=> ( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V2x),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),f2757(A_27a,V2x,V1k)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) ) ) ) ).
tff(lamtp_f2758,type,
f2758: ( del * $i * $i ) > $i ).
tff(lameq_f2758,axiom,
! [A_27a: del,V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3y: $i] :
( mem(V3y,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V5n: $i] : ( ap(f2758(A_27a,V1x,V3y),V5n) = ap(ap(c_2Erealax_2Ereal__add,ap(V1x,V5n)),ap(V3y,V5n)) ) ) ) ).
tff(conj_thm_2Enets_2ENET__ADD,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Enets_2Edorder(A_27a),V0g))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x0: tp__ty_2Erealax_2Ereal,V3y: $i] :
( mem(V3y,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V4y0: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),inj__ty_2Erealax_2Ereal(V2x0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
& p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V3y),inj__ty_2Erealax_2Ereal(V4y0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) )
=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),f2758(A_27a,V1x,V3y)),ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V2x0)),inj__ty_2Erealax_2Ereal(V4y0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) ) ) ) ) ) ).
tff(lamtp_f2759,type,
f2759: ( del * $i ) > $i ).
tff(lameq_f2759,axiom,
! [A_27a: del,V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3n: $i] : ( ap(f2759(A_27a,V1x),V3n) = ap(c_2Erealax_2Ereal__neg,ap(V1x,V3n)) ) ) ).
tff(conj_thm_2Enets_2ENET__NEG,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Enets_2Edorder(A_27a),V0g))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),inj__ty_2Erealax_2Ereal(V2x0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
<=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),f2759(A_27a,V1x)),ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V2x0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) ) ) ) ) ).
tff(lamtp_f2760,type,
f2760: ( del * $i * $i ) > $i ).
tff(lameq_f2760,axiom,
! [A_27a: del,V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3y: $i] :
( mem(V3y,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V5n: $i] : ( ap(f2760(A_27a,V1x,V3y),V5n) = ap(ap(c_2Ereal_2Ereal__sub,ap(V1x,V5n)),ap(V3y,V5n)) ) ) ) ).
tff(conj_thm_2Enets_2ENET__SUB,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Enets_2Edorder(A_27a),V0g))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x0: tp__ty_2Erealax_2Ereal,V3y: $i] :
( mem(V3y,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V4y0: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),inj__ty_2Erealax_2Ereal(V2x0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
& p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V3y),inj__ty_2Erealax_2Ereal(V4y0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) )
=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),f2760(A_27a,V1x,V3y)),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V2x0)),inj__ty_2Erealax_2Ereal(V4y0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) ) ) ) ) ) ).
tff(lamtp_f2761,type,
f2761: ( del * $i * $i ) > $i ).
tff(lameq_f2761,axiom,
! [A_27a: del,V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2y: $i] :
( mem(V2y,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V5n: $i] : ( ap(f2761(A_27a,V1x,V2y),V5n) = ap(ap(c_2Erealax_2Ereal__mul,ap(V1x,V5n)),ap(V2y,V5n)) ) ) ) ).
tff(conj_thm_2Enets_2ENET__MUL,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Enets_2Edorder(A_27a),V0g))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2y: $i] :
( mem(V2y,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3x0: tp__ty_2Erealax_2Ereal,V4y0: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),inj__ty_2Erealax_2Ereal(V3x0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
& p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V2y),inj__ty_2Erealax_2Ereal(V4y0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) )
=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),f2761(A_27a,V1x,V2y)),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V3x0)),inj__ty_2Erealax_2Ereal(V4y0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) ) ) ) ) ) ).
tff(conj_thm_2Enets_2ENET__INV,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Enets_2Edorder(A_27a),V0g))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),inj__ty_2Erealax_2Ereal(V2x0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
& ( V2x0 != surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) )
=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),f2754(A_27a,V1x)),ap(c_2Erealax_2Einv,inj__ty_2Erealax_2Ereal(V2x0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) ) ) ) ) ).
tff(lamtp_f2762,type,
f2762: ( del * $i * $i ) > $i ).
tff(lameq_f2762,axiom,
! [A_27a: del,V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3y: $i] :
( mem(V3y,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V5n: $i] : ( ap(f2762(A_27a,V1x,V3y),V5n) = ap(ap(c_2Ereal_2E_2F,ap(V1x,V5n)),ap(V3y,V5n)) ) ) ) ).
tff(conj_thm_2Enets_2ENET__DIV,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Enets_2Edorder(A_27a),V0g))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x0: tp__ty_2Erealax_2Ereal,V3y: $i] :
( mem(V3y,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V4y0: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),inj__ty_2Erealax_2Ereal(V2x0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
& p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V3y),inj__ty_2Erealax_2Ereal(V4y0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
& ( V4y0 != surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) )
=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),f2762(A_27a,V1x,V3y)),ap(ap(c_2Ereal_2E_2F,inj__ty_2Erealax_2Ereal(V2x0)),inj__ty_2Erealax_2Ereal(V4y0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) ) ) ) ) ) ).
tff(lamtp_f2763,type,
f2763: ( del * $i ) > $i ).
tff(lameq_f2763,axiom,
! [A_27a: del,V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3n: $i] : ( ap(f2763(A_27a,V1x),V3n) = ap(c_2Ereal_2Eabs,ap(V1x,V3n)) ) ) ).
tff(conj_thm_2Enets_2ENET__ABS,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x0: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),inj__ty_2Erealax_2Ereal(V2x0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),f2763(A_27a,V1x)),ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V2x0))),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g))) ) ) ) ).
tff(conj_thm_2Enets_2ENET__LE,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Enets_2Edorder(A_27a),V0g))
=> ! [V1x: $i] :
( mem(V1x,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x0: tp__ty_2Erealax_2Ereal,V3y: $i] :
( mem(V3y,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V4y0: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V1x),inj__ty_2Erealax_2Ereal(V2x0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
& p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,A_27a),V3y),inj__ty_2Erealax_2Ereal(V4y0)),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(A_27a,arr(A_27a,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0g)))
& ? [V5N: $i] :
( mem(V5N,A_27a)
& p(ap(ap(V0g,V5N),V5N))
& ! [V6n: $i] :
( mem(V6n,A_27a)
=> ( p(ap(ap(V0g,V6n),V5N))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1x,V6n)),ap(V3y,V6n))) ) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2x0)),inj__ty_2Erealax_2Ereal(V4y0))) ) ) ) ) ) ).
%------------------------------------------------------------------------------