ITP001 Axioms: ITP128+5.ax
%------------------------------------------------------------------------------
% File : ITP128+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 : lim+2.ax [Gau20]
% : HL4128+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 101 ( 5 unt; 0 def)
% Number of atoms : 713 ( 66 equ)
% Maximal formula atoms : 23 ( 7 avg)
% Number of connectives : 625 ( 13 ~; 0 |; 163 &)
% ( 11 <=>; 438 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 10 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 3 ( 2 usr; 0 prp; 1-2 aty)
% Number of functors : 66 ( 66 usr; 30 con; 0-3 aty)
% Number of variables : 389 ( 362 !; 27 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2Elim_2Econtl,axiom,
mem(c_2Elim_2Econtl,arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,bool))) ).
fof(mem_c_2Elim_2Edifferentiable,axiom,
mem(c_2Elim_2Edifferentiable,arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,bool))) ).
fof(mem_c_2Elim_2Ediffl,axiom,
mem(c_2Elim_2Ediffl,arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool)))) ).
fof(mem_c_2Elim_2Etends__real__real,axiom,
mem(c_2Elim_2Etends__real__real,arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool)))) ).
fof(ax_thm_2Elim_2Etends__real__real,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2x0] :
( mem(V2x0,ty_2Erealax_2Ereal)
=> ( p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V1l),V2x0))
<=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1l),ap(ap(c_2Epair_2E_2C(ty_2Etopology_2Etopology(ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),ap(c_2Enets_2Etendsto(ty_2Erealax_2Ereal),ap(ap(c_2Epair_2E_2C(ty_2Emetric_2Emetric(ty_2Erealax_2Ereal),ty_2Erealax_2Ereal),c_2Emetric_2Emr1),V2x0))))) ) ) ) ) ).
fof(conj_thm_2Elim_2ELIM,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1y0] :
( mem(V1y0,ty_2Erealax_2Ereal)
=> ! [V2x0] :
( mem(V2x0,ty_2Erealax_2Ereal)
=> ( p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V1y0),V2x0))
<=> ! [V3e] :
( mem(V3e,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e))
=> ? [V4d] :
( mem(V4d,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d))
& ! [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V5x),V2x0))))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V5x),V2x0))),V4d)) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V5x)),V1y0))),V3e)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2ELIM__CONST,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> p(ap(ap(ap(c_2Elim_2Etends__real__real,k(ty_2Erealax_2Ereal,V0k)),V0k),V1x)) ) ) ).
fof(lameq_f2807,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V5x] : ap(f2807(V0f,V1g),V5x) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V5x)),ap(V1g,V5x)) ) ) ).
fof(conj_thm_2Elim_2ELIM__ADD,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3m] :
( mem(V3m,ty_2Erealax_2Ereal)
=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V2l),V4x))
& p(ap(ap(ap(c_2Elim_2Etends__real__real,V1g),V3m),V4x)) )
=> p(ap(ap(ap(c_2Elim_2Etends__real__real,f2807(V0f,V1g)),ap(ap(c_2Erealax_2Ereal__add,V2l),V3m)),V4x)) ) ) ) ) ) ) ).
fof(lameq_f2808,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V5x] : ap(f2808(V0f,V1g),V5x) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V5x)),ap(V1g,V5x)) ) ) ).
fof(conj_thm_2Elim_2ELIM__MUL,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3m] :
( mem(V3m,ty_2Erealax_2Ereal)
=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V2l),V4x))
& p(ap(ap(ap(c_2Elim_2Etends__real__real,V1g),V3m),V4x)) )
=> p(ap(ap(ap(c_2Elim_2Etends__real__real,f2808(V0f,V1g)),ap(ap(c_2Erealax_2Ereal__mul,V2l),V3m)),V4x)) ) ) ) ) ) ) ).
fof(lameq_f2809,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f2809(V0f),V3x) = ap(c_2Erealax_2Ereal__neg,ap(V0f,V3x)) ) ).
fof(conj_thm_2Elim_2ELIM__NEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V1l),V2x))
<=> p(ap(ap(ap(c_2Elim_2Etends__real__real,f2809(V0f)),ap(c_2Erealax_2Ereal__neg,V1l)),V2x)) ) ) ) ) ).
fof(lameq_f2810,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f2810(V0f),V3x) = ap(c_2Erealax_2Einv,ap(V0f,V3x)) ) ).
fof(conj_thm_2Elim_2ELIM__INV,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V1l),V2x))
& V1l != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> p(ap(ap(ap(c_2Elim_2Etends__real__real,f2810(V0f)),ap(c_2Erealax_2Einv,V1l)),V2x)) ) ) ) ) ).
fof(lameq_f2811,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V5x] : ap(f2811(V0f,V1g),V5x) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V5x)),ap(V1g,V5x)) ) ) ).
fof(conj_thm_2Elim_2ELIM__SUB,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3m] :
( mem(V3m,ty_2Erealax_2Ereal)
=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V2l),V4x))
& p(ap(ap(ap(c_2Elim_2Etends__real__real,V1g),V3m),V4x)) )
=> p(ap(ap(ap(c_2Elim_2Etends__real__real,f2811(V0f,V1g)),ap(ap(c_2Ereal_2Ereal__sub,V2l),V3m)),V4x)) ) ) ) ) ) ) ).
fof(lameq_f2812,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V5x] : ap(f2812(V0f,V1g),V5x) = ap(ap(c_2Ereal_2E_2F,ap(V0f,V5x)),ap(V1g,V5x)) ) ) ).
fof(conj_thm_2Elim_2ELIM__DIV,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3m] :
( mem(V3m,ty_2Erealax_2Ereal)
=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V2l),V4x))
& p(ap(ap(ap(c_2Elim_2Etends__real__real,V1g),V3m),V4x))
& V3m != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> p(ap(ap(ap(c_2Elim_2Etends__real__real,f2812(V0f,V1g)),ap(ap(c_2Ereal_2E_2F,V2l),V3m)),V4x)) ) ) ) ) ) ) ).
fof(lameq_f2813,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V3x] : ap(f2813(V0f,V1l),V3x) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V3x)),V1l) ) ) ).
fof(conj_thm_2Elim_2ELIM__NULL,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V1l),V2x))
<=> p(ap(ap(ap(c_2Elim_2Etends__real__real,f2813(V0f,V1l)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2x)) ) ) ) ) ).
fof(conj_thm_2Elim_2ELIM__X,axiom,
! [V0x0] :
( mem(V0x0,ty_2Erealax_2Ereal)
=> p(ap(ap(ap(c_2Elim_2Etends__real__real,i(ty_2Erealax_2Ereal)),V0x0),V0x0)) ) ).
fof(conj_thm_2Elim_2ELIM__UNIQ,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2m] :
( mem(V2m,ty_2Erealax_2Ereal)
=> ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V1l),V3x))
& p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V2m),V3x)) )
=> V1l = V2m ) ) ) ) ) ).
fof(conj_thm_2Elim_2ELIM__EQUAL,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3x0] :
( mem(V3x0,ty_2Erealax_2Ereal)
=> ( ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( V4x != V3x0
=> ap(V0f,V4x) = ap(V1g,V4x) ) )
=> ( p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V2l),V3x0))
<=> p(ap(ap(ap(c_2Elim_2Etends__real__real,V1g),V2l),V3x0)) ) ) ) ) ) ) ).
fof(lameq_f2814,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V4x] : ap(f2814(V0f,V1g),V4x) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V4x)),ap(V1g,V4x)) ) ) ).
fof(conj_thm_2Elim_2ELIM__TRANSFORM,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x0] :
( mem(V2x0,ty_2Erealax_2Ereal)
=> ! [V3l] :
( mem(V3l,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Etends__real__real,f2814(V0f,V1g)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2x0))
& p(ap(ap(ap(c_2Elim_2Etends__real__real,V1g),V3l),V2x0)) )
=> p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),V3l),V2x0)) ) ) ) ) ) ).
fof(lameq_f2815,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V3h] : ap(f2815(V0f,V2x),V3h) = ap(ap(c_2Ereal_2E_2F,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,ap(ap(c_2Erealax_2Ereal__add,V2x),V3h))),ap(V0f,V2x))),V3h) ) ) ).
fof(ax_thm_2Elim_2Ediffl,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V1l),V2x))
<=> p(ap(ap(ap(c_2Elim_2Etends__real__real,f2815(V0f,V2x)),V1l),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ).
fof(lameq_f2816,axiom,
! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2h] : ap(f2816(V1x,V0f),V2h) = ap(V0f,ap(ap(c_2Erealax_2Ereal__add,V1x),V2h)) ) ) ).
fof(ax_thm_2Elim_2Econtl,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Elim_2Econtl,V0f),V1x))
<=> p(ap(ap(ap(c_2Elim_2Etends__real__real,f2816(V1x,V0f)),ap(V0f,V1x)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ).
fof(ax_thm_2Elim_2Edifferentiable,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Elim_2Edifferentiable,V0f),V1x))
<=> ? [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
& p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V1x)) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__UNIQ,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2m] :
( mem(V2m,ty_2Erealax_2Ereal)
=> ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V1l),V3x))
& p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2m),V3x)) )
=> V1l = V2m ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__CONT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V1l),V2x))
=> p(ap(ap(c_2Elim_2Econtl,V0f),V2x)) ) ) ) ) ).
fof(conj_thm_2Elim_2ECONTL__LIM,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Elim_2Econtl,V0f),V1x))
<=> p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),ap(V0f,V1x)),V1x)) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__CARAT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V1l),V2x))
<=> ? [V3g] :
( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
& ! [V4z] :
( mem(V4z,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V4z)),ap(V0f,V2x)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V3g,V4z)),ap(ap(c_2Ereal_2Ereal__sub,V4z),V2x)) )
& p(ap(ap(c_2Elim_2Econtl,V3g),V2x))
& ap(V3g,V2x) = V1l ) ) ) ) ) ).
fof(conj_thm_2Elim_2ECONT__CONST,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> p(ap(ap(c_2Elim_2Econtl,k(ty_2Erealax_2Ereal,V0k)),V1x)) ) ) ).
fof(lameq_f2817,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f2817(V0f,V1g),V3x) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ).
fof(conj_thm_2Elim_2ECONT__ADD,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Elim_2Econtl,V0f),V2x))
& p(ap(ap(c_2Elim_2Econtl,V1g),V2x)) )
=> p(ap(ap(c_2Elim_2Econtl,f2817(V0f,V1g)),V2x)) ) ) ) ) ).
fof(lameq_f2818,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f2818(V0f,V1g),V3x) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ).
fof(conj_thm_2Elim_2ECONT__MUL,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Elim_2Econtl,V0f),V2x))
& p(ap(ap(c_2Elim_2Econtl,V1g),V2x)) )
=> p(ap(ap(c_2Elim_2Econtl,f2818(V0f,V1g)),V2x)) ) ) ) ) ).
fof(lameq_f2819,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] : ap(f2819(V0f),V2x) = ap(c_2Erealax_2Ereal__neg,ap(V0f,V2x)) ) ).
fof(conj_thm_2Elim_2ECONT__NEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Elim_2Econtl,V0f),V1x))
=> p(ap(ap(c_2Elim_2Econtl,f2819(V0f)),V1x)) ) ) ) ).
fof(lameq_f2820,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] : ap(f2820(V0f),V2x) = ap(c_2Erealax_2Einv,ap(V0f,V2x)) ) ).
fof(conj_thm_2Elim_2ECONT__INV,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Elim_2Econtl,V0f),V1x))
& ap(V0f,V1x) != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> p(ap(ap(c_2Elim_2Econtl,f2820(V0f)),V1x)) ) ) ) ).
fof(lameq_f2821,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f2821(V0f,V1g),V3x) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ).
fof(conj_thm_2Elim_2ECONT__SUB,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Elim_2Econtl,V0f),V2x))
& p(ap(ap(c_2Elim_2Econtl,V1g),V2x)) )
=> p(ap(ap(c_2Elim_2Econtl,f2821(V0f,V1g)),V2x)) ) ) ) ) ).
fof(lameq_f2822,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f2822(V0f,V1g),V3x) = ap(ap(c_2Ereal_2E_2F,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ).
fof(conj_thm_2Elim_2ECONT__DIV,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Elim_2Econtl,V0f),V2x))
& p(ap(ap(c_2Elim_2Econtl,V1g),V2x))
& ap(V1g,V2x) != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> p(ap(ap(c_2Elim_2Econtl,f2822(V0f,V1g)),V2x)) ) ) ) ) ).
fof(lameq_f2823,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f2823(V0f,V1g),V3x) = ap(V1g,ap(V0f,V3x)) ) ) ).
fof(conj_thm_2Elim_2ECONT__COMPOSE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Elim_2Econtl,V0f),V2x))
& p(ap(ap(c_2Elim_2Econtl,V1g),ap(V0f,V2x))) )
=> p(ap(ap(c_2Elim_2Econtl,f2823(V0f,V1g)),V2x)) ) ) ) ) ).
fof(conj_thm_2Elim_2EIVT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ! [V3y] :
( mem(V3y,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2b))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V1a)),V3y))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3y),ap(V0f,V2b)))
& ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V4x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V4x),V2b)) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),V4x)) ) ) )
=> ? [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V5x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V2b))
& ap(V0f,V5x) = V3y ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EIVT2,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ! [V3y] :
( mem(V3y,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2b))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V2b)),V3y))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3y),ap(V0f,V1a)))
& ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V4x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V4x),V2b)) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),V4x)) ) ) )
=> ? [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V5x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V2b))
& ap(V0f,V5x) = V3y ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__CONST,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> p(ap(ap(ap(c_2Elim_2Ediffl,k(ty_2Erealax_2Ereal,V0k)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1x)) ) ) ).
fof(conj_thm_2Elim_2EDIFF__ADD,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3m] :
( mem(V3m,ty_2Erealax_2Ereal)
=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V4x))
& p(ap(ap(ap(c_2Elim_2Ediffl,V1g),V3m),V4x)) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2807(V0f,V1g)),ap(ap(c_2Erealax_2Ereal__add,V2l),V3m)),V4x)) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__MUL,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3m] :
( mem(V3m,ty_2Erealax_2Ereal)
=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V4x))
& p(ap(ap(ap(c_2Elim_2Ediffl,V1g),V3m),V4x)) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2808(V0f,V1g)),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V2l),ap(V1g,V4x))),ap(ap(c_2Erealax_2Ereal__mul,V3m),ap(V0f,V4x)))),V4x)) ) ) ) ) ) ) ).
fof(lameq_f2824,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V4x] : ap(f2824(V0f,V1c),V4x) = ap(ap(c_2Erealax_2Ereal__mul,V1c),ap(V0f,V4x)) ) ) ).
fof(conj_thm_2Elim_2EDIFF__CMUL,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V3x))
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2824(V0f,V1c)),ap(ap(c_2Erealax_2Ereal__mul,V1c),V2l)),V3x)) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__NEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V1l),V2x))
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2809(V0f)),ap(c_2Erealax_2Ereal__neg,V1l)),V2x)) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__SUB,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3m] :
( mem(V3m,ty_2Erealax_2Ereal)
=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V4x))
& p(ap(ap(ap(c_2Elim_2Ediffl,V1g),V3m),V4x)) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2811(V0f,V1g)),ap(ap(c_2Ereal_2Ereal__sub,V2l),V3m)),V4x)) ) ) ) ) ) ) ).
fof(lameq_f2825,axiom,
! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V5x] : ap(f2825(V1g,V0f),V5x) = ap(V0f,ap(V1g,V5x)) ) ) ).
fof(conj_thm_2Elim_2EDIFF__CHAIN,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3m] :
( mem(V3m,ty_2Erealax_2Ereal)
=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),ap(V1g,V4x)))
& p(ap(ap(ap(c_2Elim_2Ediffl,V1g),V3m),V4x)) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2825(V1g,V0f)),ap(ap(c_2Erealax_2Ereal__mul,V2l),V3m)),V4x)) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__X,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> p(ap(ap(ap(c_2Elim_2Ediffl,i(ty_2Erealax_2Ereal)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x)) ) ).
fof(lameq_f2826,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V2x] : ap(f2826(V0n),V2x) = ap(ap(c_2Ereal_2Epow,V2x),V0n) ) ).
fof(conj_thm_2Elim_2EDIFF__POW,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2826(V0n)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V0n)),ap(ap(c_2Ereal_2Epow,V1x),ap(ap(c_2Earithmetic_2E_2D,V0n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))),V1x)) ) ) ).
fof(lameq_f2827,axiom,
! [V1x] : ap(f2827,V1x) = ap(c_2Erealax_2Einv,V1x) ).
fof(conj_thm_2Elim_2EDIFF__XM1,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ( V0x != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2827),ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Ereal_2Epow,ap(c_2Erealax_2Einv,V0x)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),V0x)) ) ) ).
fof(conj_thm_2Elim_2EDIFF__INV,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V1l),V2x))
& ap(V0f,V2x) != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2810(V0f)),ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Ereal_2E_2F,V1l),ap(ap(c_2Ereal_2Epow,ap(V0f,V2x)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))))),V2x)) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__DIV,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3m] :
( mem(V3m,ty_2Erealax_2Ereal)
=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V4x))
& p(ap(ap(ap(c_2Elim_2Ediffl,V1g),V3m),V4x))
& ap(V1g,V4x) != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2812(V0f,V1g)),ap(ap(c_2Ereal_2E_2F,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Erealax_2Ereal__mul,V2l),ap(V1g,V4x))),ap(ap(c_2Erealax_2Ereal__mul,V3m),ap(V0f,V4x)))),ap(ap(c_2Ereal_2Epow,ap(V1g,V4x)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),V4x)) ) ) ) ) ) ) ).
fof(lameq_f2828,axiom,
! [V5r] :
( mem(V5r,ty_2Enum_2Enum)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))
=> ! [V6x] : ap(f2828(V5r,V0f),V6x) = ap(ap(V0f,V5r),V6x) ) ) ).
fof(lameq_f2829,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))
=> ! [V7x] :
( mem(V7x,ty_2Erealax_2Ereal)
=> ! [V8n] : ap(f2829(V0f,V7x),V8n) = ap(ap(V0f,V8n),V7x) ) ) ).
fof(lameq_f2830,axiom,
! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))
=> ! [V7x] : ap(f2830(V2m,V3n,V0f),V7x) = ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),V2m),V3n)),f2829(V0f,V7x)) ) ) ) ).
fof(lameq_f2831,axiom,
! [V1f_27] :
( mem(V1f_27,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))
=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ! [V9r] : ap(f2831(V1f_27,V4x),V9r) = ap(ap(V1f_27,V9r),V4x) ) ) ).
fof(conj_thm_2Elim_2EDIFF__SUM,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))
=> ! [V1f_27] :
( mem(V1f_27,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ! [V5r] :
( mem(V5r,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V5r))
& p(ap(ap(c_2Eprim__rec_2E_3C,V5r),ap(ap(c_2Earithmetic_2E_2B,V2m),V3n))) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2828(V5r,V0f)),ap(ap(V1f_27,V5r),V4x)),V4x)) ) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2830(V2m,V3n,V0f)),ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),V2m),V3n)),f2831(V1f_27,V4x))),V4x)) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2ECONT__BOUNDED,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2b))
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),V3x)) ) ) )
=> ? [V4M] :
( mem(V4M,ty_2Erealax_2Ereal)
& ! [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V5x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V2b)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V5x)),V4M)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2ECONT__HASSUP,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2b))
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),V3x)) ) ) )
=> ? [V4M] :
( mem(V4M,ty_2Erealax_2Ereal)
& ! [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V5x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V2b)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V5x)),V4M)) ) )
& ! [V6N] :
( mem(V6N,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V6N),V4M))
=> ? [V7x] :
( mem(V7x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V7x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V7x),V2b))
& p(ap(ap(c_2Erealax_2Ereal__lt,V6N),ap(V0f,V7x))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2ECONT__ATTAINS,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2b))
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),V3x)) ) ) )
=> ? [V4M] :
( mem(V4M,ty_2Erealax_2Ereal)
& ! [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V5x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V2b)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V5x)),V4M)) ) )
& ? [V6x] :
( mem(V6x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V6x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V6x),V2b))
& ap(V0f,V6x) = V4M ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2ECONT__ATTAINS2,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2b))
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),V3x)) ) ) )
=> ? [V4M] :
( mem(V4M,ty_2Erealax_2Ereal)
& ! [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V5x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V2b)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,V4M),ap(V0f,V5x))) ) )
& ? [V6x] :
( mem(V6x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V6x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V6x),V2b))
& ap(V0f,V6x) = V4M ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2ECONT__ATTAINS__ALL,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2b))
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),V3x)) ) ) )
=> ? [V4L] :
( mem(V4L,ty_2Erealax_2Ereal)
& ? [V5M] :
( mem(V5M,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ereal_2Ereal__lte,V4L),V5M))
& ! [V6y] :
( mem(V6y,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V4L),V6y))
& p(ap(ap(c_2Ereal_2Ereal__lte,V6y),V5M)) )
=> ? [V7x] :
( mem(V7x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V7x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V7x),V2b))
& ap(V0f,V7x) = V6y ) ) )
& ! [V8x] :
( mem(V8x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V8x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V8x),V2b)) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V4L),ap(V0f,V8x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V8x)),V5M)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__LINC,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V1x))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2l)) )
=> ? [V3d] :
( mem(V3d,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d))
& ! [V4h] :
( mem(V4h,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4h))
& p(ap(ap(c_2Erealax_2Ereal__lt,V4h),V3d)) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(V0f,V1x)),ap(V0f,ap(ap(c_2Erealax_2Ereal__add,V1x),V4h)))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__LDEC,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V1x))
& p(ap(ap(c_2Erealax_2Ereal__lt,V2l),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) )
=> ? [V3d] :
( mem(V3d,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d))
& ! [V4h] :
( mem(V4h,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4h))
& p(ap(ap(c_2Erealax_2Ereal__lt,V4h),V3d)) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(V0f,V1x)),ap(V0f,ap(ap(c_2Ereal_2Ereal__sub,V1x),V4h)))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__LMAX,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V1x))
& ? [V3d] :
( mem(V3d,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d))
& ! [V4y] :
( mem(V4y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V1x),V4y))),V3d))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V4y)),ap(V0f,V1x))) ) ) ) )
=> V2l = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__LMIN,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V1x))
& ? [V3d] :
( mem(V3d,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d))
& ! [V4y] :
( mem(V4y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V1x),V4y))),V3d))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V1x)),ap(V0f,V4y))) ) ) ) )
=> V2l = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__LCONST,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V1x))
& ? [V3d] :
( mem(V3d,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d))
& ! [V4y] :
( mem(V4y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V1x),V4y))),V3d))
=> ap(V0f,V4y) = ap(V0f,V1x) ) ) ) )
=> V2l = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(conj_thm_2Elim_2EINTERVAL__LEMMA,axiom,
! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ! [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V2x))
& p(ap(ap(c_2Erealax_2Ereal__lt,V2x),V1b)) )
=> ? [V3d] :
( mem(V3d,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d))
& ! [V4y] :
( mem(V4y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V2x),V4y))),V3d))
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V0a),V4y))
& p(ap(ap(c_2Ereal_2Ereal__lte,V4y),V1b)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EROLLE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V2b))
& ap(V0f,V1a) = ap(V0f,V2b)
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),V3x)) ) )
& ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V4x))
& p(ap(ap(c_2Erealax_2Ereal__lt,V4x),V2b)) )
=> p(ap(ap(c_2Elim_2Edifferentiable,V0f),V4x)) ) ) )
=> ? [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V5z))
& p(ap(ap(c_2Erealax_2Ereal__lt,V5z),V2b))
& p(ap(ap(ap(c_2Elim_2Ediffl,V0f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5z)) ) ) ) ) ) ).
fof(lameq_f2832,axiom,
! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f2832(V1a,V2b,V0f),V3x) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V3x)),ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2E_2F,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V2b)),ap(V0f,V1a))),ap(ap(c_2Ereal_2Ereal__sub,V2b),V1a))),V3x)) ) ) ) ).
fof(lameq_f2833,axiom,
! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V4x] : ap(f2833(V1a,V2b,V0f),V4x) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V4x)),ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2E_2F,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V2b)),ap(V0f,V1a))),ap(ap(c_2Ereal_2Ereal__sub,V2b),V1a))),V4x)) ) ) ) ).
fof(conj_thm_2Elim_2EMVT__LEMMA,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ap(f2832(V1a,V2b,V0f),V1a) = ap(f2833(V1a,V2b,V0f),V2b) ) ) ) ).
fof(conj_thm_2Elim_2EMVT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V2b))
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),V3x)) ) )
& ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V4x))
& p(ap(ap(c_2Erealax_2Ereal__lt,V4x),V2b)) )
=> p(ap(ap(c_2Elim_2Edifferentiable,V0f),V4x)) ) ) )
=> ? [V5l] :
( mem(V5l,ty_2Erealax_2Ereal)
& ? [V6z] :
( mem(V6z,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V6z))
& p(ap(ap(c_2Erealax_2Ereal__lt,V6z),V2b))
& p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V5l),V6z))
& ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V2b)),ap(V0f,V1a)) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,V2b),V1a)),V5l) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__ISCONST__END,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V2b))
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),V3x)) ) )
& ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V4x))
& p(ap(ap(c_2Erealax_2Ereal__lt,V4x),V2b)) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,V0f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4x)) ) ) )
=> ap(V0f,V2b) = ap(V0f,V1a) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__ISCONST,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V2b))
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),V3x)) ) )
& ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V4x))
& p(ap(ap(c_2Erealax_2Ereal__lt,V4x),V2b)) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,V0f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4x)) ) ) )
=> ! [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V5x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V2b)) )
=> ap(V0f,V5x) = ap(V0f,V1a) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__ISCONST__ALL,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> p(ap(ap(ap(c_2Elim_2Ediffl,V0f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1x)) )
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V3y] :
( mem(V3y,ty_2Erealax_2Ereal)
=> ap(V0f,V2x) = ap(V0f,V3y) ) ) ) ) ).
fof(conj_thm_2Elim_2EINTERVAL__ABS,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Erealax_2Ereal)
=> ! [V2d] :
( mem(V2d,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2Ereal__sub,V0x),V2d)),V1z))
& p(ap(ap(c_2Ereal_2Ereal__lte,V1z),ap(ap(c_2Erealax_2Ereal__add,V0x),V2d))) )
<=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V1z),V0x))),V2d)) ) ) ) ) ).
fof(conj_thm_2Elim_2ECONT__INJ__LEMMA,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V3d] :
( mem(V3d,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d))
& ! [V4z] :
( mem(V4z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V4z),V2x))),V3d))
=> ap(V1g,ap(V0f,V4z)) = V4z ) )
& ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V5z),V2x))),V3d))
=> p(ap(ap(c_2Elim_2Econtl,V0f),V5z)) ) ) )
=> ~ ! [V6z] :
( mem(V6z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V6z),V2x))),V3d))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V6z)),ap(V0f,V2x))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2ECONT__INJ__LEMMA2,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V3d] :
( mem(V3d,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d))
& ! [V4z] :
( mem(V4z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V4z),V2x))),V3d))
=> ap(V1g,ap(V0f,V4z)) = V4z ) )
& ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V5z),V2x))),V3d))
=> p(ap(ap(c_2Elim_2Econtl,V0f),V5z)) ) ) )
=> ~ ! [V6z] :
( mem(V6z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V6z),V2x))),V3d))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V2x)),ap(V0f,V6z))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2ECONT__INJ__RANGE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V3d] :
( mem(V3d,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d))
& ! [V4z] :
( mem(V4z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V4z),V2x))),V3d))
=> ap(V1g,ap(V0f,V4z)) = V4z ) )
& ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V5z),V2x))),V3d))
=> p(ap(ap(c_2Elim_2Econtl,V0f),V5z)) ) ) )
=> ? [V6e] :
( mem(V6e,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V6e))
& ! [V7y] :
( mem(V7y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V7y),ap(V0f,V2x)))),V6e))
=> ? [V8z] :
( mem(V8z,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V8z),V2x))),V3d))
& ap(V0f,V8z) = V7y ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2ECONT__INVERSE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V3d] :
( mem(V3d,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d))
& ! [V4z] :
( mem(V4z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V4z),V2x))),V3d))
=> ap(V1g,ap(V0f,V4z)) = V4z ) )
& ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V5z),V2x))),V3d))
=> p(ap(ap(c_2Elim_2Econtl,V0f),V5z)) ) ) )
=> p(ap(ap(c_2Elim_2Econtl,V1g),ap(V0f,V2x))) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__INVERSE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ! [V4d] :
( mem(V4d,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d))
& ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V5z),V3x))),V4d))
=> ap(V1g,ap(V0f,V5z)) = V5z ) )
& ! [V6z] :
( mem(V6z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V6z),V3x))),V4d))
=> p(ap(ap(c_2Elim_2Econtl,V0f),V6z)) ) )
& p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V3x))
& V2l != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,V1g),ap(c_2Erealax_2Einv,V2l)),ap(V0f,V3x))) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__INVERSE__LT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ! [V4d] :
( mem(V4d,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d))
& ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V5z),V3x))),V4d))
=> ap(V1g,ap(V0f,V5z)) = V5z ) )
& ! [V6z] :
( mem(V6z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V6z),V3x))),V4d))
=> p(ap(ap(c_2Elim_2Econtl,V0f),V6z)) ) )
& p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V3x))
& V2l != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,V1g),ap(c_2Erealax_2Einv,V2l)),ap(V0f,V3x))) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EINTERVAL__CLEMMA,axiom,
! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ! [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V2x))
& p(ap(ap(c_2Erealax_2Ereal__lt,V2x),V1b)) )
=> ? [V3d] :
( mem(V3d,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d))
& ! [V4y] :
( mem(V4y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V4y),V2x))),V3d))
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V4y))
& p(ap(ap(c_2Erealax_2Ereal__lt,V4y),V1b)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elim_2EDIFF__INVERSE__OPEN,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3a] :
( mem(V3a,ty_2Erealax_2Ereal)
=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ! [V5b] :
( mem(V5b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V3a),V4x))
& p(ap(ap(c_2Erealax_2Ereal__lt,V4x),V5b))
& ! [V6z] :
( mem(V6z,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V3a),V6z))
& p(ap(ap(c_2Erealax_2Ereal__lt,V6z),V5b)) )
=> ( ap(V1g,ap(V0f,V6z)) = V6z
& p(ap(ap(c_2Elim_2Econtl,V0f),V6z)) ) ) )
& p(ap(ap(ap(c_2Elim_2Ediffl,V0f),V2l),V4x))
& V2l != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,V1g),ap(c_2Erealax_2Einv,V2l)),ap(V0f,V4x))) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------